-
question_answer1)
sIf \[f(x)=\cos (\log x)\], then \[f(x)f(y)-\frac{1}{2}[f(x/y)+f(xy)]=\] [IIT 1983; RPET 1995; MP PET 1995; Karnataka CET 1999; UPSEAT 2001]
A)
\[-1\] done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[-2\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer2)
If \[f(x)=\frac{1-x}{1+x},\] the n \[f[f(\cos \ 2\theta )]=\] [MP PET 1994, 2001; Pb. CET 2002]
A)
\[\tan 2\theta \] done
clear
B)
\[\sec 2\theta \] done
clear
C)
\[\cos 2\theta \] done
clear
D)
\[\cot 2\theta \] done
clear
View Solution play_arrow
-
question_answer3)
If \[f(x)=\sin \log x\], then the value of \[f(xy)+f\left( \frac{x}{y} \right)-2f(x).\cos \log y\] is equal to [Orissa JEE 2004]
A)
1 done
clear
B)
0 done
clear
C)
?1 done
clear
D)
\[\sin \log x.\cos \log y\] done
clear
View Solution play_arrow
-
question_answer4)
The value of b and c for which the identity \[f(x+1)-f(x)=8x+3\] is satisfied, where \[f(x)=b{{x}^{2}}+cx+d\], are [Roorkee 1992]
A)
\[b=2,\ c=1\] done
clear
B)
\[b=4,\ c=-1\] done
clear
C)
\[b=-1,\ c=4\] done
clear
D)
\[b=-1,\ c=1\] done
clear
View Solution play_arrow
-
question_answer5)
Given the function \[f(x)=\frac{{{a}^{x}}+{{a}^{-x}}}{2},\ (a>2)\]. Then \[f(x+y)+f(x-y)=\]
A)
\[2f(x).f(y)\] done
clear
B)
\[f(x).f(y)\] done
clear
C)
\[\frac{f(x)}{f(y)}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer6)
If \[f(x)=\frac{x}{x-1}\], then \[\frac{f(a)}{f(a+1)}=\] [MP PET 1996]
A)
\[f(-a)\] done
clear
B)
\[f\left( \frac{1}{a} \right)\] done
clear
C)
\[f({{a}^{2}})\] done
clear
D)
\[f\left( \frac{-a}{a-1} \right)\] done
clear
View Solution play_arrow
-
question_answer7)
If \[f(x)=\cos (\log x)\], then \[f({{x}^{2}})f({{y}^{2}})-\frac{1}{2}\left[ f\,\left( \frac{{{x}^{2}}}{2} \right)+f\left( \frac{{{x}^{2}}}{{{y}^{2}}} \right) \right]\] has the value [MNR 1992]
A)
?2 done
clear
B)
?1 done
clear
C)
½ done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer8)
The equivalent function of \[\log {{x}^{2}}\] is [MP PET 1997]
A)
\[2\log x\] done
clear
B)
\[2\log |x|\] done
clear
C)
\[|\log {{x}^{2}}|\] done
clear
D)
\[{{(\log x)}^{2}}\] done
clear
View Solution play_arrow
-
question_answer9)
If \[f(x)=\log \left[ \frac{1+x}{1-x} \right]\], then \[f\left[ \frac{2x}{1+{{x}^{2}}} \right]\] is equal to [MP PET 1999; RPET 1999; UPSEAT 2003]
A)
\[{{[f(x)]}^{2}}\] done
clear
B)
\[{{[f(x)]}^{3}}\] done
clear
C)
\[2f(x)\] done
clear
D)
\[3f(x)\] done
clear
View Solution play_arrow
-
question_answer10)
If \[\varphi (x)={{a}^{x}}\], then \[{{\{\varphi (p)\}}^{3}}\]is equal to [MP PET 1999]
A)
\[\varphi (3p)\] done
clear
B)
\[3\varphi (p)\] done
clear
C)
\[6\varphi (p)\] done
clear
D)
\[2\varphi (p)\] done
clear
View Solution play_arrow
-
question_answer11)
If \[f(x)=\frac{x-3}{x+1}\], then \[f[f\{f(x)\}]\] equals [RPET 1996]
A)
x done
clear
B)
?x done
clear
C)
\[\frac{x}{2}\] done
clear
D)
\[-\frac{1}{x}\] done
clear
View Solution play_arrow
-
question_answer12)
If \[f(x)=\cos (\log x)\], then the value of \[f(x).f(4)-\frac{1}{2}\left[ f\left( \frac{x}{4} \right)+f(4x) \right]\] [Kurukshetra CEE 1998]
A)
1 done
clear
B)
?1 done
clear
C)
0 done
clear
D)
\[\pm 1\] done
clear
View Solution play_arrow
-
question_answer13)
If \[f(x)=\frac{x-|x|}{|x|}\], then \[f(-1)=\] [SCRA 1996]
A)
1 done
clear
B)
?2 done
clear
C)
0 done
clear
D)
+2 done
clear
View Solution play_arrow
-
question_answer14)
If \[f(x)=4{{x}^{3}}+3{{x}^{2}}+3x+4\], then \[{{x}^{3}}f\left( \frac{1}{x} \right)\] is [SCRA 1996]
A)
\[f(-x)\] done
clear
B)
\[\frac{1}{f(x)}\] done
clear
C)
\[{{\left( f\left( \frac{1}{x} \right) \right)}^{2}}\] done
clear
D)
\[f(x)\] done
clear
View Solution play_arrow
-
question_answer15)
Let \[f:R\to R\] be defined by \[f(x)=2x+|x|\], then \[f(2x)+f(-x)-f(x)=\] [EAMCET 2000]
A)
\[2x\] done
clear
B)
\[2|x|\] done
clear
C)
\[-2x\] done
clear
D)
\[-2|x|\] done
clear
View Solution play_arrow
-
question_answer16)
If \[f(x+ay,\ x-ay)=axy\], then \[f(x,\ y)\] is equal to [AMU 2001]
A)
xy done
clear
B)
\[{{x}^{2}}-{{a}^{2}}{{y}^{2}}\] done
clear
C)
\[\frac{{{x}^{2}}-{{y}^{2}}}{4}\] done
clear
D)
\[\frac{{{x}^{2}}-{{y}^{2}}}{{{a}^{2}}}\] done
clear
View Solution play_arrow
-
question_answer17)
If \[f(x)=\cos [{{\pi }^{2}}]x+\cos [-{{\pi }^{2}}]x\], then [Orissa JEE 2002]
A)
\[f\left( \frac{\pi }{4} \right)=2\] done
clear
B)
\[f(-\pi )=2\] done
clear
C)
\[f(\pi )=1\] done
clear
D)
\[f\left( \frac{\pi }{2} \right)=-1\] done
clear
View Solution play_arrow
-
question_answer18)
If \[f(x)=\frac{1}{\sqrt{x+2\sqrt{2x-4}}}+\frac{1}{\sqrt{x-2\sqrt{2x-4}}}\] for \[x>2\], then \[f(11)=\] [EAMCET 2003]
A)
7/6 done
clear
B)
5/6 done
clear
C)
6/7 done
clear
D)
5/7 done
clear
View Solution play_arrow
-
question_answer19)
If \[{{e}^{f(x)}}=\frac{10+x}{10-x},\ x\in (-10,\ 10)\] and \[f(x)=kf\left( \frac{200x}{100+{{x}^{2}}} \right)\], then \[k=\] [EAMCET 2003]
A)
0.5 done
clear
B)
0.6 done
clear
C)
0.7 done
clear
D)
0.8 done
clear
View Solution play_arrow
-
question_answer20)
If \[f(x)=2\sin x\], \[g(x)={{\cos }^{2}}x\], then \[(f+g)\left( \frac{\pi }{3} \right)=\]
A)
1 done
clear
B)
\[\frac{2\sqrt{3}+1}{4}\] done
clear
C)
\[\sqrt{3}+\frac{1}{4}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer21)
The graph of the function \[y=f(x)\] is symmetrical about the line \[x=2\], then [AIEEE 2004]
A)
\[f(x)=-f(-x)\] done
clear
B)
\[f(2+x)=f(2-x)\] done
clear
C)
\[f(x)=f(-x)\] done
clear
D)
\[f(x+2)=f(x-2)\] done
clear
View Solution play_arrow
-
question_answer22)
If \[f(x)=\frac{x}{x-1}=\frac{1}{y}\], then \[f(y)=\] [MP PET 1995, 97]
A)
x done
clear
B)
\[x+1\] done
clear
C)
\[x-1\] done
clear
D)
\[1-x\] done
clear
View Solution play_arrow
-
question_answer23)
If \[y=f(x)=\frac{ax+b}{cx-a}\], then x is equal to [AMU 2001]
A)
\[1/f(x)\] done
clear
B)
\[1/f(y)\] done
clear
C)
\[yf(x)\] done
clear
D)
\[f(y)\] done
clear
View Solution play_arrow
-
question_answer24)
If \[f(x)=\frac{{{x}^{2}}-1}{{{x}^{2}}+1}\], for every real numbers. then the minimum value of f [Pb. CET 2001]
A)
Does not exist because f is bounded done
clear
B)
Is not attained even through f is bounded done
clear
C)
Is equal to +1 done
clear
D)
Is equal to ?1 done
clear
View Solution play_arrow
-
question_answer25)
\[f(x,\ y)=\frac{1}{x+y}\] is a homogeneous function of degree [Orissa JEE 2004]
A)
1 done
clear
B)
?1 done
clear
C)
2 done
clear
D)
?2 done
clear
View Solution play_arrow
-
question_answer26)
Let x be a non-zero rational number and y be an irrational number. Then xy is [Orissa JEE 2004]
A)
Rational done
clear
B)
Irrational done
clear
C)
Non-zero done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer27)
Numerical value of the expression \[\left| \ \frac{3{{x}^{3}}+1}{2{{x}^{2}}+2}\ \right|\] for \[x=-3\] is [Orissa JEE 2004; UPSEAT 2004]
A)
4 done
clear
B)
2 done
clear
C)
3 done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer28)
The function \[f:R\to R,\ f(x)={{x}^{2}},\forall x\in R\] is [MP PET 1997]
A)
Injection but not surjection done
clear
B)
Surjection but not injection done
clear
C)
Injection as well as surjection done
clear
D)
Neither injection nor surjection done
clear
View Solution play_arrow
-
question_answer29)
If for two functions g and f, gof is both injective and surjective, then which of the following is true [Kurukshetra CEE 1998]
A)
g and f should be injective and surjective done
clear
B)
g should be injective and surjective done
clear
C)
f should be injective and surjective done
clear
D)
None of them may be surjective and injective done
clear
View Solution play_arrow
-
question_answer30)
The function which map [?1, 1] to [0, 2] are [Kurukshetra CEE 1998]
A)
One linear function done
clear
B)
Two linear function done
clear
C)
Circular function done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer31)
Let \[f(x)=\left\{ \begin{align} & \frac{1}{2},\ if\ 0\le x\le \frac{1}{2} \\ & \frac{1}{3},\ if\ \frac{1}{2}<x\le 1 \\ \end{align} \right.\], then f is [SCRA 1996]
A)
A rational function done
clear
B)
A trigonometric function done
clear
C)
A step function done
clear
D)
An exponential function done
clear
View Solution play_arrow
-
question_answer32)
Function \[f:R\to R,\ f(x)={{x}^{2}}+x\] is [RPET 1999]
A)
One-one onto done
clear
B)
One-one into done
clear
C)
Many-one onto done
clear
D)
Many-one into done
clear
View Solution play_arrow
-
question_answer33)
Mapping \[f:R\to R\] which is defined as \[f(x)=\cos x,\ x\in R\] will be [UPSEAT 1999]
A)
Neither one-one nor onto done
clear
B)
One-one done
clear
C)
Onto done
clear
D)
One-one onto done
clear
View Solution play_arrow
-
question_answer34)
The function \[f:R\to R\] defined by \[f(x)=(x-1)\] \[(x-2)(x-3)\] is [Roorkee 1999]
A)
One-one but not onto done
clear
B)
Onto but not one-one done
clear
C)
Both one-one and onto done
clear
D)
Neither one-one nor onto done
clear
View Solution play_arrow
-
question_answer35)
. If \[f:R\to R\], then \[f(x)=\ |x|\] is [RPET 2000]
A)
One-one but not onto done
clear
B)
Onto but not one-one done
clear
C)
One-one and onto done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer36)
Which of the four statements given below is different from others [UPSEAT 2000]
A)
\[f:A\to B\] done
clear
B)
\[f:x\to f(x)\] done
clear
C)
f is a mapping of A into B done
clear
D)
f is a function of A into B done
clear
View Solution play_arrow
-
question_answer37)
Let \[f:N\to N\] defined by \[f(x)={{x}^{2}}+x+1\], \[x\in N\], then f is [AMU 2000]
A)
One-one onto done
clear
B)
Many one onto done
clear
C)
One-one but not onto done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer38)
Let X and Y be subsets of R, the set of all real numbers. The function \[f:X\to Y\]defined by \[f(x)={{x}^{2}}\] for \[x\in X\] is one-one but not onto if (Here \[{{R}^{+}}\] is the set of all positive real numbers) [EAMCET 2000]
A)
\[X=Y={{R}^{+}}\] done
clear
B)
\[X=R,\ Y={{R}^{+}}\] done
clear
C)
\[X={{R}^{+}},\ Y=R\] done
clear
D)
\[X=Y=R\] done
clear
View Solution play_arrow
-
question_answer39)
Set A has 3 elements and set B has 4 elements. The number of injection that can be defined from A to B is [UPSEAT 2001]
A)
144 done
clear
B)
12 done
clear
C)
24 done
clear
D)
64 done
clear
View Solution play_arrow
-
question_answer40)
Let \[f:R\to R\] be a function defined by \[f(x)=\frac{x-m}{x-n}\], where \[m\ne n\]. Then [UPSEAT 2001]
A)
f is one-one onto done
clear
B)
f is one-one into done
clear
C)
f is many one onto done
clear
D)
f is many one into done
clear
View Solution play_arrow
-
question_answer41)
The function \[f:R\to R\] defined by \[f(x)={{e}^{x}}\] is [Karnataka CET 2002; UPSEAT 2002]
A)
Onto done
clear
B)
Many-one done
clear
C)
One-one and into done
clear
D)
Many one and onto done
clear
View Solution play_arrow
-
question_answer42)
Which one of the following is a objective function on the set of real numbers [Kerala (Engg.) 2002]
A)
\[2x-5\] done
clear
B)
\[|x|\] done
clear
C)
\[{{x}^{2}}\] done
clear
D)
\[{{x}^{2}}+1\] done
clear
View Solution play_arrow
-
question_answer43)
Let \[f(x)=\frac{{{x}^{2}}-4}{{{x}^{2}}+4}\] for \[|x|\ >2\], then the function \[f:(-\infty ,\ -2]\cup [2,\ \infty )\to (-1,\ 1)\] is [Orissa JEE 2002]
A)
One-one into done
clear
B)
One-one onto done
clear
C)
Many one into done
clear
D)
Many one onto done
clear
View Solution play_arrow
-
question_answer44)
Let the function \[f:R\to R\] be defined by \[f(x)=2x+\sin x,\ x\in R\]. Then f is [IIT Screening 2002]
A)
One-to-one and onto done
clear
B)
One-to-one but not onto done
clear
C)
Onto but not one-to-one done
clear
D)
Neither one-to-one nor onto done
clear
View Solution play_arrow
-
question_answer45)
A function f from the set of natural numbers to integers defined by \[f(n)=\left\{ \begin{align} & \frac{n-1}{2},\ \text{when}\ n\ \text{is}\ \text{odd} \\ & -\frac{n}{2},\ \text{when }n\text{ is even} \\ \end{align} \right.\], is [AIEEE 2003]
A)
One-one but not onto done
clear
B)
Onto but not one-one done
clear
C)
One-one and onto both done
clear
D)
Neither one-one nor onto done
clear
View Solution play_arrow
-
question_answer46)
If \[f:[0,\ \infty )\to [0,\ \infty )\] and \[f(x)=\frac{x}{1+x},\]then f is [IIT Screening 2003]
A)
One-one and onto done
clear
B)
One-one but not onto done
clear
C)
Onto but not one-one done
clear
D)
Neither one-one nor onto done
clear
View Solution play_arrow
-
question_answer47)
If \[f:R\to S\] defined by \[f(x)=\sin x-\sqrt{3}\cos x+1\]is onto, then the interval of S is [AIEEE 2004; IIT Screening 2004]
A)
[?1, 3] done
clear
B)
[1, 1] done
clear
C)
[0, 1] done
clear
D)
[0, ?1] done
clear
View Solution play_arrow
-
question_answer48)
If R denotes the set of all real numbers then the function \[f:R\to R\] defined \[f(x)=\ [x]\] [Karnataka CET 2004]
A)
One-one only done
clear
B)
Onto only done
clear
C)
Both one-one and onto done
clear
D)
Neither one-one nor onto done
clear
View Solution play_arrow
-
question_answer49)
\[f(x)=x+\sqrt{{{x}^{2}}}\] is a function from R\[\to \]R , then \[f(x)\] is [Orissa JEE 2004]
A)
Injective done
clear
B)
Surjective done
clear
C)
Bijective done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer50)
If \[(x,\,y)\in R\] and \[x,\ y\ne 0\]; \[f(x,\ y)\to \frac{x}{y}\], then this function is a/an [Orissa JEE 2004]
A)
Surjection done
clear
B)
Bijection done
clear
C)
One-one done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer51)
The period of \[f(x)=x-[x]\], if it is periodic, is [AMU 2000]
A)
\[f(x)\] is not periodic done
clear
B)
\[\frac{1}{2}\] done
clear
C)
1 done
clear
D)
2 done
clear
View Solution play_arrow
-
question_answer52)
If \[f(x)\] is periodic function with period T then the function \[f(ax+b)\] where \[a>0\], is periodic with period [AMU 2000]
A)
\[T/b\] done
clear
B)
aT done
clear
C)
bT done
clear
D)
\[T/a\] done
clear
View Solution play_arrow
-
question_answer53)
If \[f(x)=ax+b\] and \[g(x)=cx+d\], then \[f(g(x))=g(f(x))\] is equivalent to [UPSEAT 2001]
A)
\[f(a)=g(c)\] done
clear
B)
\[f(b)=g(b)\] done
clear
C)
\[f(d)=g(b)\] done
clear
D)
\[f(c)=g(a)\] done
clear
View Solution play_arrow
-
question_answer54)
Domain and range of \[f(x)=\frac{|x-3|}{x-3}\] are respectively
A)
\[R,\ [-1,\ 1]\] done
clear
B)
\[R-\{3\},\ \left\{ 1,\ -1 \right\}\] done
clear
C)
\[{{R}^{+}},\ R\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer55)
If in greatest integer function, the domain is a set of real numbers, then range will be set of
A)
Real numbers done
clear
B)
Rational numbers done
clear
C)
Imaginary numbers done
clear
D)
Integers done
clear
View Solution play_arrow
-
question_answer56)
Domain of function \[f(x)={{\sin }^{-1}}5x\] is
A)
\[\left( -\frac{1}{5},\ \frac{1}{5} \right)\] done
clear
B)
\[\left[ -\frac{1}{5},\ \frac{1}{5} \right]\] done
clear
C)
R done
clear
D)
\[\left( 0,\ \frac{1}{5} \right)\] done
clear
View Solution play_arrow
-
question_answer57)
The domain of the function \[f(x)=\frac{{{\sin }^{-1}}(3-x)}{\ln (|x|\ -2)}\] is [Orissa JEE 2002]
A)
[2, 4] done
clear
B)
(2, 3) È (3, 4] done
clear
C)
[2,\[\infty \]) done
clear
D)
\[(-\infty ,\ -3)\cup [2,\ \infty )\] done
clear
View Solution play_arrow
-
question_answer58)
The domain of \[{{\sin }^{-1}}\left[ {{\log }_{3}}\left( \frac{x}{3} \right) \right]\] is [AIEEE 2002]
A)
[1, 9] done
clear
B)
[?1, 9] done
clear
C)
[?9, 1] done
clear
D)
[?9, ?1] done
clear
View Solution play_arrow
-
question_answer59)
Domain of the function \[\log |{{x}^{2}}-9|\] is
A)
R done
clear
B)
\[R-[-3,\ 3]\] done
clear
C)
\[R-\{-3,\ 3\}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer60)
Domain of \[f(x)=\log |\log x|\] is [DCE 2002]
A)
\[(0,\ \infty )\] done
clear
B)
\[(1,\ \infty )\] done
clear
C)
\[(0,\ 1)\cup (1,\ \infty )\] done
clear
D)
\[(-\infty ,\ 1)\] done
clear
View Solution play_arrow
-
question_answer61)
The domain of the function \[f(x)={{\sin }^{-1}}[{{\log }_{2}}(x/2)]\] is [RPET 2002]
A)
[1, 4] done
clear
B)
[?4, 1] done
clear
C)
[?1, 4] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer62)
The domain of \[f(x)=\frac{{{\log }_{2}}(x+3)}{{{x}^{2}}+3x+2}\] is [IIT Screening 2001; UPSEAT 2001]
A)
\[R-\{-1,\ -2\}\] done
clear
B)
\[(-2,\ +\infty )\] done
clear
C)
\[R-\{-1,\ -2,\ -3\}\] done
clear
D)
\[(-3,\ +\infty )-\{-1,\ -2\}\] done
clear
View Solution play_arrow
-
question_answer63)
The function \[f(x)=\frac{{{\sec }^{-1}}x}{\sqrt{x-[x]}},\] where [.] denotes the greatest integer less than or equal to x is defined for all x belonging to
A)
R done
clear
B)
\[R-\{(-1,\ 1)\cup (n|n\in Z)\}\] done
clear
C)
\[{{R}^{+}}-(0,\ 1)\] done
clear
D)
\[{{R}^{+}}-\{n|n\in N\}\] done
clear
View Solution play_arrow
-
question_answer64)
If the domain of function \[f(x)={{x}^{2}}-6x+7\] is \[(-\infty ,\ \infty )\], then the range of function is [MP PET 1996]
A)
\[(-\infty ,\ \infty )\] done
clear
B)
\[[-2,\ \infty )\] done
clear
C)
\[(-2,\ 3)\] done
clear
D)
\[(-\infty ,\ -2)\] done
clear
View Solution play_arrow
-
question_answer65)
The domain of the function \[f(x)=\sqrt{\log \frac{1}{|\sin x|}}\] is [RPET 2001]
A)
\[R-\{2n\pi ,\ n\in I\}\] done
clear
B)
\[R-\{n\pi ,\ n\in I\}\] done
clear
C)
\[R-\{-\pi ,\ \pi \}\] done
clear
D)
\[(-\infty ,\ \infty )\] done
clear
View Solution play_arrow
-
question_answer66)
The domain of the function \[f(x)=\log (\sqrt{x-4}+\sqrt{6-x})\] is [RPET 2001]
A)
\[[4,\infty )\] done
clear
B)
\[(-\infty ,\ 6]\] done
clear
C)
\[[4,\ 6]\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer67)
. Domain of the function \[f(x)={{\left[ {{\log }_{10}}\left( \frac{5x-{{x}^{2}}}{4} \right) \right]}^{1/2}}\] is [UPSEAT 2001]
A)
\[-\infty <x<\infty \] done
clear
B)
\[1\le x\le 4\] done
clear
C)
\[4\le x\le 16\] done
clear
D)
\[-1\le x\le 1\] done
clear
View Solution play_arrow
-
question_answer68)
The domain of the derivative of the function \[f(x)=\left\{ \begin{align} & {{\tan }^{-1}}x\ \ \ \ \ ,\ |x|\ \le 1 \\ & \frac{1}{2}(|x|\ -1)\ ,\ |x|\ >1 \\ \end{align} \right.\] is [IIT Screening 2002]
A)
\[R-\{0\}\] done
clear
B)
\[R-\{1\}\] done
clear
C)
\[R-\{-1\}\] done
clear
D)
\[R-\{-1,\ 1\}\] done
clear
View Solution play_arrow
-
question_answer69)
The domain of the function \[f(x)={{\log }_{3+x}}({{x}^{2}}-1)\] is [Orissa JEE 2003]
A)
\[(-3,\ -1)\cup (1,\ \infty )\] done
clear
B)
\[[-3,\ -1)\cup [1,\ \infty )\] done
clear
C)
\[(-3,\ -2)\cup (-2,\ -1)\cup (1,\ \infty )\] done
clear
D)
\[[-3,\ -2)\cup (-2,\ -1)\cup [1,\ \infty )\] done
clear
View Solution play_arrow
-
question_answer70)
If ?n? is an integer, the domain of the function \[\sqrt{\sin 2x}\] is [MP PET 2003]
A)
\[\left[ n\pi -\frac{\pi }{2},\ n\pi \right]\] done
clear
B)
\[\left[ n\pi ,\ n\pi +\frac{\pi }{2} \right]\] done
clear
C)
\[[(2n-1)\pi ,\ 2n\pi ]\] done
clear
D)
\[[2n\pi ,\ (2n+1)\pi ]\] done
clear
View Solution play_arrow
-
question_answer71)
Domain of definition of the function \[f(x)=\frac{3}{4-{{x}^{2}}}+{{\log }_{10}}({{x}^{3}}-x)\], is [AIEEE 2003]
A)
(1, 2) done
clear
B)
\[(-1,\ 0)\cup (1,\ 2)\] done
clear
C)
\[(1,\ 2)\cup (2,\ \infty )\] done
clear
D)
\[(-1,\ 0)\cup (1,\ 2)\cup (2,\ \infty )\] done
clear
View Solution play_arrow
-
question_answer72)
Domain of the function \[f(x)=\sqrt{2-2x-{{x}^{2}}}\] is [BIT Ranchi 1992]
A)
\[-\sqrt{3}\le x\le \sqrt{3}\] done
clear
B)
\[-1-\sqrt{3}\le x\le -1+\sqrt{3}\] done
clear
C)
\[-2\le x\le 2\] done
clear
D)
\[-2+\sqrt{3}\le x\le -2-\sqrt{3}\] done
clear
View Solution play_arrow
-
question_answer73)
Domain of the function \[f(x)=\frac{x-3}{(x-1)\sqrt{{{x}^{2}}-4}}\] is [BIT Ranchi 1991]
A)
(1, 2) done
clear
B)
\[(-\infty ,\ -2)\cup (2,\ \infty )\] done
clear
C)
\[(-\infty ,\ -2)\cup (1,\ \infty )\] done
clear
D)
\[(-\infty ,\ \infty )-\{1,\ \pm 2\}\] done
clear
View Solution play_arrow
-
question_answer74)
Domain of the function \[\sqrt{\log \left\{ (5x-{{x}^{2}})/6 \right\}}\] is
A)
(2, 3) done
clear
B)
[2, 3] done
clear
C)
[1, 2] done
clear
D)
[1, 3] done
clear
View Solution play_arrow
-
question_answer75)
Domain of the function \[\sqrt{2-x}-\frac{1}{\sqrt{9-{{x}^{2}}}}\] is
A)
(?3, 1) done
clear
B)
[?3, 1] done
clear
C)
(?3, 2] done
clear
D)
[?3, 1) done
clear
View Solution play_arrow
-
question_answer76)
Domain of the function \[\frac{\sqrt{1+x}-\sqrt{1-x}}{x}\] is
A)
(?1, 1) done
clear
B)
(?1, 1)?{0} done
clear
C)
[?1, 1] done
clear
D)
[?1, 1]?{0} done
clear
View Solution play_arrow
-
question_answer77)
The domain of the function \[f(x)=\sqrt{x-{{x}^{2}}}+\sqrt{4+x}+\sqrt{4-x}\] is [AMU 1999]
A)
\[[-4,\ \infty )\] done
clear
B)
[?4, 4] done
clear
C)
[0, 4] done
clear
D)
[0, 1] done
clear
View Solution play_arrow
-
question_answer78)
The domain of the function \[f(x)={{\sin }^{-1}}\{{{(1+{{e}^{x}})}^{-1}}\}\] is [AMU 1999]
A)
\[\left( \frac{1}{4},\ \frac{1}{3} \right)\] done
clear
B)
[?1, 0] done
clear
C)
[0, 1] done
clear
D)
[?1, 1] done
clear
View Solution play_arrow
-
question_answer79)
The domain of the function \[\sqrt{\log ({{x}^{2}}-6x+6)}\] is [Roorkee 1999; MP PET 2002]
A)
\[(-\infty ,\ \infty )\] done
clear
B)
\[(-\infty ,\ 3-\sqrt{3})\cup (3+\sqrt{3},\ \infty )\] done
clear
C)
\[(-\infty ,\ 1]\cup [5,\ \infty )\] done
clear
D)
\[[0,\ \infty )\] done
clear
View Solution play_arrow
-
question_answer80)
The largest possible set of real numbers which can be the domain of \[f(x)=\sqrt{1-\frac{1}{x}}\] is [AMU 2000]
A)
\[(0,\ 1)\cup (0,\ \infty )\] done
clear
B)
\[(-1,\ 0)\cup (1,\ \infty )\] done
clear
C)
\[(-\infty ,\ -1)\cup (0,\ \infty )\] done
clear
D)
\[(-\infty ,\ 0)\cup (1,\ \infty )\] done
clear
View Solution play_arrow
-
question_answer81)
Domain of the function \[f(x)={{\sin }^{-1}}(1+3x+2{{x}^{2}})\] is [Roorkee 2000]
A)
\[(-\infty ,\ \infty )\] done
clear
B)
\[(-1,\ 1)\] done
clear
C)
\[\left[ -\frac{3}{2},\ 0 \right]\] done
clear
D)
\[\left( -\infty ,\ \frac{-1}{2} \right)\cup (2,\ \infty )\] done
clear
View Solution play_arrow
-
question_answer82)
Domain of the function \[f(x)=\frac{{{x}^{2}}-3x+2}{{{x}^{2}}+x-6}\] is
A)
\[\{x:x\in R,\ \ x\ne 3\}\] done
clear
B)
\[\{x:x\in R,\ \ x\ne 2\}\] done
clear
C)
\[\{x:x\in R\}\] done
clear
D)
\[\{x:x\in R,\ \ x\ne 2,\ x\ne -3\}\] done
clear
View Solution play_arrow
-
question_answer83)
Domain of \[f(x)={{({{x}^{2}}-1)}^{-1/2}}\] is [Roorkee 1987]
A)
\[(-\infty ,\ -1)\cup (1,\ \infty )\] done
clear
B)
\[(-\infty ,\ -1]\cup (1,\ \infty )\] done
clear
C)
\[(-\infty ,\ -1]\cup [1,\ \infty )\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer84)
The domain of the function \[y=\frac{1}{\sqrt{|x|\ -x}}\] is [Roorkee 1998; RPET 2000]
A)
\[(-\infty ,\ 0)\] done
clear
B)
\[(-\infty ,\ 0]\] done
clear
C)
\[(-\infty ,\ -1)\] done
clear
D)
\[(-\infty ,\ \infty )\] done
clear
View Solution play_arrow
-
question_answer85)
The natural domain of the real valued function defined by \[f(x)=\sqrt{{{x}^{2}}-1}+\sqrt{{{x}^{2}}+1}\] is [SCRA 1996]
A)
\[1<x<\infty \] done
clear
B)
\[-\infty <x<\infty \] done
clear
C)
\[-\infty <x<-1\] done
clear
D)
\[(-\infty ,\ \infty )-(-1,\ 1)\] done
clear
View Solution play_arrow
-
question_answer86)
The domain of the function \[f(x)=\exp (\sqrt{5x-3-2{{x}^{2}}})\] is [MP PET 2004]
A)
\[\left[ 1,\ -\frac{3}{2} \right]\] done
clear
B)
\[\left[ \frac{3}{2},\ \infty \right]\] done
clear
C)
\[[-\infty ,\ 1]\] done
clear
D)
\[\left[ 1,\ \frac{3}{2} \right]\] done
clear
View Solution play_arrow
-
question_answer87)
The domain of the function \[f(x)=\frac{{{\sin }^{-1}}(x-3)}{\sqrt{9-{{x}^{2}}}}\] is [AIEEE 2004]
A)
[1, 2) done
clear
B)
[2, 3) done
clear
C)
[1, 2] done
clear
D)
[2, 3] done
clear
View Solution play_arrow
-
question_answer88)
The range of \[f(x)=\sec \left( \frac{\pi }{4}{{\cos }^{2}}x \right)\,,\ -\infty <x<\infty \] is [Orissa JEE 2002]
A)
\[[1,\ \sqrt{2}]\] done
clear
B)
\[[1,\ \infty )\] done
clear
C)
\[[-\sqrt{2},\ -1]\cup [1,\ \sqrt{2}]\] done
clear
D)
\[(-\infty ,\ -1]\cup [1,\ \infty )\] done
clear
View Solution play_arrow
-
question_answer89)
Range of the function \[f(x)=\frac{{{x}^{2}}+x+2}{{{x}^{2}}+x+1};\ x\in R\] is [IIT Screening 2003]
A)
\[(1,\ \infty )\] done
clear
B)
\[(1,\ 11/7]\] done
clear
C)
\[(1,\ 7/3]\] done
clear
D)
\[(1,\ 7/5]\] done
clear
View Solution play_arrow
-
question_answer90)
If \[f(x)=a\cos (bx+c)+d\], then range of \[f(x)\] is [UPSEAT 2001]
A)
\[[d+a,\ d+2a]\] done
clear
B)
\[[a-d,\ a+d]\] done
clear
C)
\[[d+a,\ a-d]\] done
clear
D)
\[[d-a,\ d+a]\] done
clear
View Solution play_arrow
-
question_answer91)
Range of \[f(x)=\ [x]\ -x\] is
A)
[0, 1] done
clear
B)
(?1, 0] done
clear
C)
R done
clear
D)
(?1, 1) done
clear
View Solution play_arrow
-
question_answer92)
The range of \[f(x)=\cos (x/3)\] is [RPET 2002]
A)
\[(-1/3,\ 1/3)\] done
clear
B)
\[[-1,\ 1]\] done
clear
C)
\[(1/3,\ -1/3)\] done
clear
D)
\[(-3,\ 3)\] done
clear
View Solution play_arrow
-
question_answer93)
The range of the function \[f(x)=\frac{x+2}{|x+2|}\] is [RPET 2002]
A)
{0, 1} done
clear
B)
{?1, 1} done
clear
C)
R done
clear
D)
\[R-\{-2\}\] done
clear
View Solution play_arrow
-
question_answer94)
The range of \[f(x)=\cos x-\sin x\] is [MP PET 1995; Pb. CET 2001]
A)
\[(-1,\ 1)\] done
clear
B)
\[[-1,\,\ 1)\] done
clear
C)
\[\left[ -\frac{\pi }{2},\ \frac{\pi }{2} \right]\] done
clear
D)
\[[-\sqrt{2},\ \sqrt{2}]\] done
clear
View Solution play_arrow
-
question_answer95)
If \[f:R\to R\], then the range of the function \[f(x)=\frac{{{x}^{2}}}{{{x}^{2}}+1}\] is [MP PET 1987]
A)
\[{{R}^{-}}\] done
clear
B)
\[{{R}^{+}}\] done
clear
C)
R done
clear
D)
\[R\times R\] done
clear
View Solution play_arrow
-
question_answer96)
The range of \[f(x)=\cos 2x-\sin 2x\] contains the set [IIT Screening]
A)
[2, 4] done
clear
B)
[?1, 1] done
clear
C)
[?2, 2] done
clear
D)
[?4, 4] done
clear
View Solution play_arrow
-
question_answer97)
Range of the function \[\frac{1}{2-\sin 3x}\] is [AMU 1999]
A)
[1, 3] done
clear
B)
\[\left[ \frac{1}{3},\,\,1 \right]\] done
clear
C)
(1, 3) done
clear
D)
\[\left( \frac{1}{3},\ 1 \right)\] done
clear
View Solution play_arrow
-
question_answer98)
Range of the function \[f(x)={{\sin }^{2}}({{x}^{4}})+{{\cos }^{2}}({{x}^{4}})\] is
A)
\[(-\infty ,\ \infty )\] done
clear
B)
{1} done
clear
C)
(?1, 1) done
clear
D)
(0, 1) done
clear
View Solution play_arrow
-
question_answer99)
Range of the function \[f(x)=9-7\sin x\] is
A)
(2, 16) done
clear
B)
[2, 16] done
clear
C)
[?1, 1] done
clear
D)
(2, 16] done
clear
View Solution play_arrow
-
question_answer100)
Range of \[f(x)=\frac{{{x}^{2}}+34x-71}{{{x}^{2}}+2x-7}\] is [Roorkee 1983]
A)
[5, 9] done
clear
B)
\[(-\infty ,\ 5]\cup [9,\ \infty )\] done
clear
C)
(5, 9) done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer101)
The interval for which \[{{\sin }^{-1}}\sqrt{x}+{{\cos }^{-1}}\sqrt{x}=\frac{\pi }{2}\] holds [IIT Screening]
A)
\[[0,\ \infty )\] done
clear
B)
\[[0,\ 3]\] done
clear
C)
[0, 1] done
clear
D)
[0, 2] done
clear
View Solution play_arrow
-
question_answer102)
Function \[{{\sin }^{-1}}\sqrt{x}\] is defined in the interval
A)
(?1, 1) done
clear
B)
[0, 1] done
clear
C)
[?1, 0] done
clear
D)
(?1, 2) done
clear
View Solution play_arrow
-
question_answer103)
The function \[f:R\to R\] is defined by \[f(x)={{\cos }^{2}}x+{{\sin }^{4}}x\] for \[x\in R\], then \[f(R)=\] [EAMCET 2002]
A)
\[\left( \frac{3}{4},\ 1 \right]\] done
clear
B)
\[\left[ \frac{3}{4},\ 1 \right)\] done
clear
C)
\[\left[ \frac{3}{4},\ 1 \right]\] done
clear
D)
\[\left( \frac{3}{4},\ 1 \right)\] done
clear
View Solution play_arrow
-
question_answer104)
If x is real, then value of the expression \[\frac{{{x}^{2}}+14x+9}{{{x}^{2}}+2x+3}\] lies between [UPSEAT 2002]
A)
5 and 4 done
clear
B)
5 and ?4 done
clear
C)
? 5 and 4 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer105)
For \[\theta >\frac{\pi }{3}\], the value of \[f(\theta )={{\sec }^{2}}\theta +{{\cos }^{2}}\theta \] always lies in the interval [Orissa JEE 2002]
A)
(0, 2) done
clear
B)
[0, 1] done
clear
C)
(1, 2) done
clear
D)
\[[2,\ \infty )\] done
clear
View Solution play_arrow
-
question_answer106)
Which of the following function is even function [RPET 2000]
A)
\[f(x)=\frac{{{a}^{x}}+1}{{{a}^{x}}-1}\] done
clear
B)
\[f(x)=x\left( \frac{{{a}^{x}}-1}{{{a}^{x}}+1} \right)\] done
clear
C)
\[f(x)=\frac{{{a}^{x}}-{{a}^{-x}}}{{{a}^{x}}+{{a}^{-x}}}\] done
clear
D)
\[f(x)=\sin x\] done
clear
View Solution play_arrow
-
question_answer107)
If \[f(x)=\log \frac{1+x}{1-x}\], then \[f(x)\] is [Kerala (Engg.) 2002]
A)
Even function done
clear
B)
\[f({{x}_{1}})f({{x}_{2}})=f({{x}_{1}}+{{x}_{2}})\] done
clear
C)
\[\frac{f({{x}_{1}})}{f({{x}_{2}})}=f({{x}_{1}}-{{x}_{2}})\] done
clear
D)
Odd function done
clear
View Solution play_arrow
-
question_answer108)
The function \[f(x)=\sin \left( \log (x+\sqrt{{{x}^{2}}+1}) \right)\] is [Orissa JEE 2002]
A)
Even function done
clear
B)
Odd function done
clear
C)
Neither even nor odd done
clear
D)
Periodic function done
clear
View Solution play_arrow
-
question_answer109)
The function \[f(x)=\log (x+\sqrt{{{x}^{2}}+1})\], is [AIEEE 2003; MP PET 2003; UPSEAT 2003]
A)
An even function done
clear
B)
An odd function done
clear
C)
A Periodic function done
clear
D)
Neither an even nor odd function done
clear
View Solution play_arrow
-
question_answer110)
Which of the following function is invertible [AMU 2001]
A)
\[f(x)={{2}^{x}}\] done
clear
B)
\[f(x)={{x}^{3}}-x\] done
clear
C)
\[f(x)={{x}^{2}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer111)
If \[y=f(x)=\frac{x+2}{x-1}\], then \[x=\] [IIT 1984]
A)
\[f(y)\] done
clear
B)
\[2f(y)\] done
clear
C)
\[\frac{1}{f(y)}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer112)
Which of the following functions is inverse of itself
A)
\[f(x)=\frac{1-x}{1+x}\] done
clear
B)
\[f(x)={{5}^{\log x}}\] done
clear
C)
\[f(x)={{2}^{x(x-1)}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer113)
The inverse of the function \[f(x)=\frac{{{e}^{x}}-{{e}^{-x}}}{{{e}^{x}}+{{e}^{-x}}}+2\] is given by [Kurukshetra CEE 1996]
A)
\[{{\log }_{e}}{{\left( \frac{x-2}{x-1} \right)}^{1/2}}\] done
clear
B)
\[{{\log }_{e}}{{\left( \frac{x-1}{3-x} \right)}^{1/2}}\] done
clear
C)
\[{{\log }_{e}}{{\left( \frac{x}{2-x} \right)}^{1/2}}\] done
clear
D)
\[{{\log }_{e}}{{\left( \frac{x-1}{x+1} \right)}^{-2}}\] done
clear
View Solution play_arrow
-
question_answer114)
If the function \[f:[1,\ \infty )\to [1,\ \infty )\] is defined by \[f(x)={{2}^{x(x-1)}},\] then \[{{f}^{-1}}\](x) is [IIT 1999]
A)
\[{{\left( \frac{1}{2} \right)}^{x(x-1)}}\] done
clear
B)
\[\frac{1}{2}(1+\sqrt{1+4{{\log }_{2}}x})\] done
clear
C)
\[\frac{1}{2}(1-\sqrt{1+4{{\log }_{2}}x})\] done
clear
D)
Not defined done
clear
View Solution play_arrow
-
question_answer115)
If \[f(x)=3x-5\], then \[{{f}^{-1}}(x)\] [IIT 1998]
A)
Is given by \[\frac{1}{3x-5}\] done
clear
B)
Is given by \[\frac{x+5}{3}\] done
clear
C)
Does not exist because f is not one-one done
clear
D)
Does not exist because f is not onto done
clear
View Solution play_arrow
-
question_answer116)
If \[f:IR\to IR\] is defined by \[f(x)=3x-4\], then \[{{f}^{-1}}:IR\to IR\] is [SCRA 1996]
A)
\[4-3x\] done
clear
B)
\[\frac{x+4}{3}\] done
clear
C)
\[\frac{1}{3x-4}\] done
clear
D)
\[\frac{3}{x+4}\] done
clear
View Solution play_arrow
-
question_answer117)
If \[f(x)=\frac{x}{1+x}\], then \[{{f}^{-1}}(x)\] is equal to [AMU 1999]
A)
\[\frac{(1+x)}{x}\] done
clear
B)
\[\frac{1}{(1+x)}\] done
clear
C)
\[\frac{(1+x)}{(1-x)}\] done
clear
D)
\[\frac{x}{(1-x)}\] done
clear
View Solution play_arrow
-
question_answer118)
Which of the following function is inverse function [AMU 2000]
A)
\[f(x)=\frac{1}{x-1}\] done
clear
B)
\[f(x)={{x}^{2}}\] for all\[x\] done
clear
C)
\[f(x)={{x}^{2}}\], \[x\ge 0\] done
clear
D)
\[f(x)={{x}^{2}},\ x\le 0\] done
clear
View Solution play_arrow
-
question_answer119)
Let \[f(\theta )=\sin \theta (\sin \theta +\sin 3\theta )\], then \[f(\theta )\] [IIT Screening 2000]
A)
\[\ge 0\] only when \[\theta \ge 0\] done
clear
B)
\[\le 0\] for all real \[\theta \] done
clear
C)
\[\ge 0\] for all real \[\theta \] done
clear
D)
\[\le 0\]only when \[\theta \le 0\] done
clear
View Solution play_arrow
-
question_answer120)
The inverse of the function \[\frac{{{10}^{x}}-{{10}^{-x}}}{{{10}^{x}}+{{10}^{-x}}}\] is [RPET 2001]
A)
\[\frac{1}{2}{{\log }_{10}}\left( \frac{1+x}{1-x} \right)\] done
clear
B)
\[\frac{1}{2}{{\log }_{10}}\left( \frac{1-x}{1+x} \right)\] done
clear
C)
\[\frac{1}{4}{{\log }_{10}}\left( \frac{2x}{2-x} \right)\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer121)
Inverse of the function \[y=2x-3\] is [UPSEAT 2002]
A)
\[\frac{x+3}{2}\] done
clear
B)
\[\frac{x-3}{2}\] done
clear
C)
\[\frac{1}{2x-3}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer122)
Let the function f be defined by \[f(x)=\frac{2x+1}{1-3x}\], then \[{{f}^{-1}}(x)\] is [Kerala (Engg.) 2002]
A)
\[\frac{x-1}{3x+2}\] done
clear
B)
\[\frac{3x+2}{x-1}\] done
clear
C)
\[\frac{x+1}{3x-2}\] done
clear
D)
\[\frac{2x+1}{1-3x}\] done
clear
View Solution play_arrow
-
question_answer123)
If \[f(x)={{x}^{2}}+1\], then \[{{f}^{-1}}(17)\] and \[{{f}^{-1}}(-3)\]will be [UPSEAT 2003]
A)
4, 1 done
clear
B)
4, 0 done
clear
C)
3, 2 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer124)
Let \[f(x)=\sin x+\cos x,\ g(x)={{x}^{2}}-1\]. Thus \[g(f(x))\] is invertible for \[x\in \] [IIT Screening 2004]
A)
\[\left[ -\frac{\pi }{2},\ 0 \right]\] done
clear
B)
\[\left[ -\frac{\pi }{2},\ \pi \right]\] done
clear
C)
\[\left[ -\frac{\pi }{2},\ \frac{\pi }{4} \right]\] done
clear
D)
\[\left[ 0,\ \frac{\pi }{2} \right]\] done
clear
View Solution play_arrow
-
question_answer125)
If \[f(x)=\frac{2x-1}{x+5}\]\[(x\ne -5)\], then \[{{f}^{-1}}(x)\] is equal to [MP PET 2004]
A)
\[\frac{x+5}{2x-1},\ x\ne \frac{1}{2}\] done
clear
B)
\[\frac{5x+1}{2-x},\ x\ne 2\] done
clear
C)
\[\frac{5x-1}{2-x},\ x\ne 2\] done
clear
D)
\[\frac{x-5}{2x+1},\ x\ne \frac{1}{2}\] done
clear
View Solution play_arrow
-
question_answer126)
If f be the greatest integer function and g be the modulus function, then \[(gof)\left( -\frac{5}{3} \right)-(fog)\left( -\frac{5}{3} \right)=\]
A)
1 done
clear
B)
?1 done
clear
C)
2 done
clear
D)
4 done
clear
View Solution play_arrow
-
question_answer127)
If \[f(x)=2x\] and g is identity function, then
A)
\[(fog)(x)=g(x)\] done
clear
B)
\[(g+g)(x)=g(x)\] done
clear
C)
\[(fog)(x)=(g+g)(x)\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer128)
If \[f(x)={{x}^{2}}-1\] and \[g(x)=3x+1\], then \[(gof)(x)=\]
A)
\[{{x}^{2}}-1\] done
clear
B)
\[2{{x}^{2}}-1\] done
clear
C)
\[3{{x}^{2}}-2\] done
clear
D)
\[2{{x}^{2}}+2\] done
clear
View Solution play_arrow
-
question_answer129)
If f is an exponential function and g is a logarithmic function, then \[fog(1)\] will be
A)
e done
clear
B)
\[{{\log }_{e}}e\] done
clear
C)
0 done
clear
D)
2e done
clear
View Solution play_arrow
-
question_answer130)
If \[f(x)={{e}^{2x}}\] and \[g(x)=\log \sqrt{x}\]\[(x>0)\], then \[fog(x)\] is equal to
A)
\[{{e}^{2x}}\] done
clear
B)
\[\log \sqrt{x}\] done
clear
C)
\[{{e}^{2x}}\log \sqrt{x}\] done
clear
D)
x done
clear
View Solution play_arrow
-
question_answer131)
If \[f(x)=|\cos x|\]and \[g(x)=[x]\], then \[gof(x)\] is equal to
A)
\[|\cos \ [x]|\] done
clear
B)
\[|\cos x|\] done
clear
C)
\[[|\cos x|]\] done
clear
D)
\[|[\cos x]|\] done
clear
View Solution play_arrow
-
question_answer132)
If \[f(x)={{x}^{2}}+1\],then \[fof(x)\] is equal to
A)
\[{{x}^{2}}+1\] done
clear
B)
\[{{x}^{2}}+2x+2\] done
clear
C)
\[{{x}^{4}}+2{{x}^{2}}+2\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer133)
If \[f(x)=\frac{x}{\sqrt{1+{{x}^{2}}}}\], then \[(fofof)(x)=\] [RPET 2000]
A)
\[\frac{3x}{\sqrt{1+{{x}^{2}}}}\] done
clear
B)
\[\frac{x}{\sqrt{1+3{{x}^{2}}}}\] done
clear
C)
\[\frac{3x}{\sqrt{1+{{x}^{2}}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer134)
If \[\varphi (x)={{x}^{2}}+1\] and \[\psi (x)={{3}^{x}}\], then \[\varphi \{\psi (x)\}\] and \[\psi \{\varphi (x)\}=\]
A)
\[{{3}^{2x+1}},\ {{3}^{{{x}^{2}}+1}}\] done
clear
B)
\[{{3}^{2x+1}},\ {{3}^{{{x}^{2}}}}+1\] done
clear
C)
\[{{3}^{2x}}+1,\ {{3}^{{{x}^{2}}+1}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer135)
If \[g(x)={{x}^{2}}+x-2\] and \[\frac{1}{2}gof(x)=2{{x}^{2}}-5x+2\], then \[f(x)\] is [Roorkee 1998; MP PET 2002]
A)
\[2x-3\] done
clear
B)
\[2x+3\] done
clear
C)
\[2{{x}^{2}}+3x+1\] done
clear
D)
\[2{{x}^{2}}-3x-1\] done
clear
View Solution play_arrow
-
question_answer136)
If \[f(x)={{\log }_{a}}x\] and \[F(x)={{a}^{x}}\], then \[F[f(x)]\] is [SCRA 1996]
A)
\[f[F(x)]\] done
clear
B)
\[f[F(2x)]\] done
clear
C)
\[F|f(2x)|\] done
clear
D)
\[F[(x)]\] done
clear
View Solution play_arrow
-
question_answer137)
Let f and g be functions defined by \[f(x)=\frac{x}{x+1},\]\[g(x)=\frac{x}{1-x}\], then \[(fog)(x)\] is [SCRA 1996]
A)
\[\frac{1}{x}\] done
clear
B)
\[\frac{1}{x-1}\] done
clear
C)
\[x-1\] done
clear
D)
x done
clear
View Solution play_arrow
-
question_answer138)
If from \[R\to R\], \[f(x)={{(x+1)}^{2}}\], \[g(x)={{x}^{2}}+1\], then \[(fog)(-3)\] equals [RPET 1999]
A)
121 done
clear
B)
112 done
clear
C)
211 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer139)
Suppose that \[g(x)=1+\sqrt{x}\] and \[f(g(x))=3+2\sqrt{x}+x\], then \[f(x)\] is [MP PET 2000; Karnataka CET 2002]
A)
\[1+2{{x}^{2}}\] done
clear
B)
\[2+{{x}^{2}}\] done
clear
C)
\[1+x\] done
clear
D)
\[2+x\] done
clear
View Solution play_arrow
-
question_answer140)
The composite mapping \[fog\]of the map \[f:R\to R\], \[f(x)=\sin x\], \[g:R\to R\], \[g(x)={{x}^{2}}\]is [UPSEAT 2000]
A)
\[\sin x+{{x}^{2}}\] done
clear
B)
\[{{(\sin x)}^{2}}\] done
clear
C)
\[\sin {{x}^{2}}\] done
clear
D)
\[\frac{\sin x}{{{x}^{2}}}\] done
clear
View Solution play_arrow
-
question_answer141)
Let \[f(x)=ax+b\] and \[g(x)=cx+d,\ a\ne 0,\ c\ne 0\]. Assume \[a=1,\ b=2\]. If \[(fog)(x)=(gof)(x)\] for all x, what can you say about c and d [AMU 2000]
A)
c and d both arbitrary done
clear
B)
\[c=1,\ d\] arbitrary done
clear
C)
c arbitrary, \[d=1\] done
clear
D)
\[c=1,\ d=1\] done
clear
View Solution play_arrow
-
question_answer142)
Let \[g(x)=1+x-[x]\] and \[f(x)=\left\{ \begin{align} & -1,\ x<0 \\ & 0,\ \ x=0,\ \\ & \text{1,}\ \ \ \text{x}>\text{0} \\ \end{align} \right.\]then for all \[x,\ f(g(x))\] is equal to [IIT Screening 2001; UPSEAT 2001]
A)
x done
clear
B)
1 done
clear
C)
\[f(x)\] done
clear
D)
\[g(x)\] done
clear
View Solution play_arrow
-
question_answer143)
If \[f(x)=\frac{\alpha \,x}{x+1},\ x\ne -1\]. Then, for what value of \[\alpha \] is \[f(f(x))=x\] [IIT Screening 2001; UPSEAT 2001]
A)
\[\sqrt{2}\] done
clear
B)
\[-\sqrt{2}\] done
clear
C)
1 done
clear
D)
?1 done
clear
View Solution play_arrow
-
question_answer144)
If \[f(x)=\frac{2x+1}{3x-2}\], then \[(fof)(2)\] is equal to [Kerala (Engg.) 2002]
A)
1 done
clear
B)
3 done
clear
C)
4 done
clear
D)
2 done
clear
View Solution play_arrow
-
question_answer145)
If \[f(x)={{\sin }^{2}}x\] and the composite function \[g\{f(x)\}=|\sin x|\], then the function \[g(x)\] is equal to [Orissa JEE 2003]
A)
\[\sqrt{x-1}\] done
clear
B)
\[\sqrt{x}\] done
clear
C)
\[\sqrt{x+1}\] done
clear
D)
\[-\sqrt{x}\] done
clear
View Solution play_arrow
-
question_answer146)
If \[f(x)={{(a-{{x}^{n}})}^{1/n}},\]where \[a>0\]and n is a positive integer, then \[f[f(x)]=\] [IIT 1983; UPSEAT 2001, 04]
A)
\[{{x}^{3}}\] done
clear
B)
\[{{x}^{2}}\] done
clear
C)
\[x\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer147)
Let \[f:(-1,1)\to B\], be a function defined by \[f(x)={{\tan }^{-1}}\frac{2x}{1-{{x}^{2}}},\] then f is both one- one and onto when B is the interval [AIEEE 2005]
A)
\[\left[ -\frac{\pi }{2},\frac{\pi }{2} \right]\] done
clear
B)
\[\left( -\frac{\pi }{2},\frac{\pi }{2} \right)\] done
clear
C)
\[\left( 0,\frac{\pi }{2} \right)\] done
clear
D)
\[\left[ 0,\frac{\pi }{2} \right)\] done
clear
View Solution play_arrow
-
question_answer148)
A real valued function \[f(x)\] satisfies the function equation \[f(x-y)=f(x)f(y)-f(a-x)f(a+y)\] where a is a given constant and \[f(0)=1\], \[f(2a-x)\] is equal to [AIEEE 2005]
A)
\[f(a)+f(a-x)\] done
clear
B)
\[f(-x)\] done
clear
C)
\[-f(x)\] done
clear
D)
\[f(x)\] done
clear
View Solution play_arrow
-
question_answer149)
If X and Y are two non- empty sets where \[f:X\to Y\]is function is defined such that \[f(c)=\left\{ f(x):x\in C \right\}\]for \[C\subseteq X\]and \[{{f}^{-1}}(D)=\{x:f(x)\in D\}\]for \[D\subseteq Y\] for any \[A\subseteq X\] and \[B\subseteq Y,\]then [IIT Screening 2005]
A)
\[{{f}^{-1}}(f(A))=A\] done
clear
B)
\[{{f}^{-1}}(f(A))=A\]only if \[f(x)=Y\] done
clear
C)
\[f({{f}^{-1}}(B))=B\] only if \[B\subseteq f(X)\] done
clear
D)
\[f({{f}^{-1}}(B))=B\] done
clear
View Solution play_arrow
-
question_answer150)
If \[f(x)=2{{x}^{6}}+3{{x}^{4}}+4{{x}^{2}}\] then \[f'(x)\] is [DCE 2005]
A)
Even function done
clear
B)
An odd function done
clear
C)
Neither even nor odd done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer151)
If \[f(x)=\frac{\alpha x}{x+1},x\ne -1\], for what value of \[\alpha \] is \[f(f(x))=x\] [Kerala (Engg.) 2005]
A)
\[\sqrt{2}\] done
clear
B)
\[-\sqrt{2}\] done
clear
C)
1 done
clear
D)
2 done
clear
E)
?1 done
clear
View Solution play_arrow
-
question_answer152)
Function \[f(x)=x-[\,],\] where [ ] shows a greatest integer. This function is [DCE 2005]
A)
A periodic function done
clear
B)
A periodic function whose period is \[\frac{1}{2}\] done
clear
C)
A periodic function whose period is 1 done
clear
D)
Not a periodic function done
clear
View Solution play_arrow
-
question_answer153)
Let \[g(x)=1+x-[x]\] and \[f(x)=\left\{ \begin{align} & -1,\,\,\,If\,\,x<0 \\ & \,\,0,\,\,\,If\,\,\,x=0, \\ & \,\,1,\,\,\,\,\,if\,\,\,\,x>0 \\ \end{align} \right.\]then for all values of x the value of \[fog(x)\] [DCE 2005]
A)
x done
clear
B)
1 done
clear
C)
\[f(x)\] done
clear
D)
\[g(x)\] done
clear
View Solution play_arrow
-
question_answer154)
If \[g:[-2,\,2]\to R\]where \[g(x)=\] \[{{x}^{3}}+\tan x+\left[ \frac{{{x}^{2}}+1}{P} \right]\]is a odd function then the value of parametric P is [DCE 2005]
A)
\[-5<P<5\] done
clear
B)
\[P<5\] done
clear
C)
\[P>5\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer155)
The Domain of function \[f(x)={{\log }_{e}}(x-[x])\] is [AMU 2005]
A)
R done
clear
B)
R-Z done
clear
C)
\[(0,+\infty )\] done
clear
D)
Z done
clear
View Solution play_arrow
-
question_answer156)
The domain of \[{{\sin }^{-1}}({{\log }_{3}}x)\] is [Kerala (Engg.) 2005]
A)
[?1, 1] done
clear
B)
[0, 1] done
clear
C)
[0, \[\infty \]] done
clear
D)
R done
clear
E)
[1/3, 3] done
clear
View Solution play_arrow
-
question_answer157)
If \[f({{x}_{1}})-f({{x}_{2}})=f\left( \frac{{{x}_{1}}-{{x}_{2}}}{1-{{x}_{1}}{{x}_{2}}} \right)\] for \[{{x}_{1}},{{x}_{2}}\in [-1,\,1]\], then \[f(x)\] is [Roorkee 1998]
A)
\[\log \frac{(1-x)}{(1+x)}\] done
clear
B)
\[{{\tan }^{-1}}\frac{(1-x)}{(1+x)}\] done
clear
C)
\[\log \frac{(1+x)}{(1-x)}\] done
clear
D)
\[{{\tan }^{-1}}\frac{(1+x)}{(1-x)}\] done
clear
View Solution play_arrow
-
question_answer158)
If equation of the curve remain unchanged by replacing x and y from ?x and ?y respectively, then the curve is
A)
Symmetric along the x-axis done
clear
B)
Symmetric along the y-axis done
clear
C)
Symmetric in opposite quadrants done
clear
D)
Symmetric along the line y =x done
clear
View Solution play_arrow
-
question_answer159)
If equation of the curve remain unchanged by replacing x and y from y and x respectively, then the curve is
A)
Symmetric along x-axis done
clear
B)
Symmetric along y-axis done
clear
C)
Symmetric along the line y = ? x done
clear
D)
Symmetric along the line y = x done
clear
View Solution play_arrow
-
question_answer160)
A condition for a function \[y=f(x)\] to have an inverse is that it should be
A)
Defined for all x done
clear
B)
Continuous everywhere done
clear
C)
Strictly monotonic and continuous in the domain done
clear
D)
An even function done
clear
View Solution play_arrow
-
question_answer161)
If\[f(x)=\left\{ \begin{align} & x,\,\,\text{when}\,x\,\text{is}\,\text{rational} \\ & 0\text{,}\,\,\text{when }x\text{ is irrational} \\ \end{align} \right.\]; \[g(x)=\left\{ \begin{align} & 0,\,\,\,\,\text{when}\,x\,\text{is}\,\text{rational} \\ & x,\,\,\,\,\text{when}\,x\,\text{is irrational} \\ \end{align} \right.\] then \[(f-g)\] is [IIT Screening 2005]
A)
One-one onto done
clear
B)
One-one not onto done
clear
C)
Not one-one but onto done
clear
D)
Not one-one not onto done
clear
View Solution play_arrow
-
question_answer162)
Range of the function \[f(x)=\frac{{{x}^{2}}}{{{x}^{2}}+1}\]is [Orissa JEE 2005]
A)
(?1, 0) done
clear
B)
(?1, 1) done
clear
C)
[0, 1) done
clear
D)
(1, 1) done
clear
View Solution play_arrow
-
question_answer163)
The function f satisfies the functional equation \[3f(x)+2f\left( \frac{x+59}{x-1} \right)=10x+30\] for all real \[x\ne 1\]. The value of \[f(7)\] is [Kerala (Engg.) 2005]
A)
8 done
clear
B)
4 done
clear
C)
?8 done
clear
D)
11 done
clear
E)
44 done
clear
View Solution play_arrow
-
question_answer164)
If \[{{e}^{x}}=y+\sqrt{1+{{y}^{2}}}\], then y = [MNR 1990, UPSEAT 2000]
A)
\[\frac{{{e}^{x}}+{{e}^{-x}}}{2}\] done
clear
B)
\[\frac{{{e}^{x}}-{{e}^{-x}}}{2}\] done
clear
C)
\[{{e}^{x}}+{{e}^{-x}}\] done
clear
D)
\[{{e}^{x}}-{{e}^{-x}}\] done
clear
View Solution play_arrow
-
question_answer165)
Let \[f:(2,\,3)\to (0,\,1)\] be defined by \[f(x)=x-[x]\] then \[{{f}^{-1}}(x)\] equals [Orissa JEE 2005]
A)
\[x-2\] done
clear
B)
\[x+1\] done
clear
C)
\[x-1\] done
clear
D)
\[x+2\] done
clear
View Solution play_arrow