Question

The domain and range of the function $f$ given by $f\left(x\right)=2-\left|x-5\right|$ is

• A. Domain $={\mathrm{ℝ}}^{+}$, Range $=(-\infty ,1]$
• B. Domain $=\mathrm{ℝ}$, Range $=(-\infty ,2]$
• C. Domain $=\mathrm{ℝ}$, Range $=\left(-\infty ,2\right)$
• D. Domain $={\mathrm{ℝ}}^{+}$, Range $=(-\infty ,2]$

Answer 1

The correct option is B

Domain $=\mathrm{ℝ}$, Range $=(-\infty ,2]$

Explanation for the correct answer:

$f\left(x\right)=2-\left|x-5\right|$

$f\left(x\right)$ is defined for all real values therefore the domain is $\mathrm{ℝ}$

$$\begin{gathered} \Rightarrow \left|x-5\right|\ge 0 \\ \Rightarrow -\left|x-5\right|\le 0 \\ \Rightarrow 2-\left|x-5\right|\le 2 \\ \Rightarrow f\left(x\right)\le 2 \end{gathered}$$

Therefore the range is $(-\infty ,2]$

Hence option (B) is the correct answer