A) \[Domain=\,{{R}^{+}},\,\,Range=(-\infty ,\,\,1]\]
B) \[Domain=\,R,\,\,Range=(-\infty ,\,\,2]\]
C) \[Domain=R,\text{ }Range=(-\infty ,\,\,2)\]
D) \[\operatorname{Domain} ={{R}^{+}}, Range=(-\infty ,\,\,2)\]
Correct Answer: B
Solution :
[b] Given \[f(x)=2-\left| x-5 \right|\] Domain of \[f(x)\] is defined for all real values of x. Since, \[\left| x-5 \right|\ge 0\Rightarrow -\left| x-5 \right|\le 0\] \[\Rightarrow 2-\left| x-5 \right|\le 2\Rightarrow f(x)\le 2\] Hence, range of \[f(x)\]is \[\left( -\infty ,2 \right]\]
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