Question

Let $f\left(x\right)$ be a function whose domain is $\left[-5,7\right]$. Let $g\left(x\right)=\left|2x+5\right|$. Then what is the domain of $\left(fog\right)\left(x\right)$.

• A. $\left[-4,1\right]$
• B. $\left[-5,1\right]$
• C. $\left[-6,1\right]$
• D. None of these

Answer 1

The correct option is C

$\left[-6,1\right]$

Explanation of correct answer :

Finding domain of $\left(fog\right)\left(x\right)$:

Given, $g\left(x\right)=\left|2x+5\right|$

Domain of $f\left(x\right)$ = $\left[-5,7\right]$

$\left(fog\right)\left(x\right)$ $=f\left(g\left(x\right)\right)$

Domain of $f\left(g\left(x\right)\right)$ means $g\left(x\right)$ will also lie in the same range as $x$.

i.e $-5\le \left|2x+5\right|\le 7$

But, $\left|2x+5\right|$ has to be positive as it is a modulus function.

$$\begin{gathered} \Rightarrow \left|2x+5\right|\le 7 \\ \Rightarrow -7\le 2x+5\le 7 \\ \Rightarrow -12\le 2x\le 2 \\ \Rightarrow -6\le x\le 1 \\ \Rightarrow x\in \left[-6,1\right] \end{gathered}$$

Hence, the domain of $\left(fog\right)\left(x\right)$ is $\left[-6,1\right]$.

Thus, the correct answer is Option(C).