# JEE Main & Advanced Mathematics Functions Domain, Co-domain and Range of Function

## Domain, Co-domain and Range of Function

Category : JEE Main & Advanced

If a function $f$ is defined from a set $f$ to set B then for $f:A\to B$ set A is called the domain of function $f$ and set $B$ is called the co-domain of function $f$. The set of all f-images of the elements of $A$ is called the range of function $f$.

In other words, we can say

Domain = All possible values of $x$ for which $f(x)$ exists.

Range   = For all values of $x,$ all possible values of $f(x)$.

(1) Methods for finding domain and range of function

(i) Domain

(a) Expression under even root (i.e., square root, fourth root etc.) $\ge 0$.  Denominator $\ne 0$.

If domain of $y=f(x)$ and $y=g\,(x)$ are ${{D}_{1}}$ and ${{D}_{2}}$ respectively then the domain of $f(x)\pm g\,(x)$ or $f(x).g\,(x)$ is ${{D}_{1}}\cap {{D}_{2}}.$

While domain of $\frac{f(x)}{g(x)}$ is ${{D}_{1}}\cap {{D}_{2}}-\{g(x)=0\}.$

Domain of $\left( \sqrt{f(x)} \right)={{D}_{1}}\cap \{x:f(x)\ge 0\}$

(ii) Range : Range of $y=f(x)$ is collection of all outputs $f(x)$ corresponding to each real number in the domain.

(a) If domain $\in$ finite number of points $\Rightarrow$ range $\in$ set of corresponding $f(x)$ values.

(b) If domain $\in R$ or $R-$ [some finite points]. Then express $x$ in terms of $y$. From this find $y$ for $x$ to be defined (i.e., find the values of $y$ for which $x$ exists).

(c) If domain $\in$ a finite interval, find the least and greatest value for range using monotonicity.

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