Question
The domain and range of the function $$f$$ given by $$f\left( x \right) = 2 - \left| {x - 5} \right|$$ is :
A.
$${\text{Domain }} = {R^ + }{\text{, Range}}\, = \left( { - \infty ,\,1} \right]$$
B.
$${\text{Domain }} = R{\text{, Range}}\, = \left( { - \infty ,\,2} \right]$$
C.
$${\text{Domain }} = R{\text{, Range}}\, = \left( { - \infty ,\,2} \right)$$
D.
$${\text{Domain }} = {R^ + }{\text{, Range}}\, = \left( { - \infty ,\,2} \right]$$
Answer :
$${\text{Domain }} = R{\text{, Range}}\, = \left( { - \infty ,\,2} \right]$$
Solution :
Given $$f\left( x \right) = 2 - \left| {x - 5} \right|$$
Domain of $$f\left( x \right)$$ is defined for all real values of $$x$$
$$\eqalign{
& {\text{Since, }}\left| {x - 5} \right| \geqslant 0 \cr
& \Rightarrow - \left| {x - 5} \right| \leqslant 0 \cr
& \Rightarrow 2 - \left| {x - 5} \right| \leqslant 2 \cr
& \Rightarrow f\left( x \right) \leqslant 2 \cr} $$
Hence, range of $$f\left( x \right)$$ is $$\left( { - \infty ,\,2} \right]$$