Question

Let $f\left( x \right)$ be a function whose domain is $\left[ -5,7 \right]$. Let $g\left( x \right)=\left| 2x+5 \right|$. The domain of $f\left( g\left( x \right) \right)$ is
A. $\left[ -4,1 \right]$
B. $\left[ -5,1 \right]$
C. $\left[ -6,1 \right]$
D. None of these

Answer 1

We first find the relation between the range of $g\left( x \right)$ and the domain of $f\left( x \right)$ for the composite function $f\left( g\left( x \right) \right)$. We use that to find the range of the function $g\left( x \right)=\left| 2x+5 \right|$. We simplify the expression to find the required solution.

Complete answer:
$f\left( g\left( x \right) \right)$ is a composite function for which the range of $g\left( x \right)$ has to be the subset of the domain of $f\left( x \right)$.
The domain of $f\left( x \right)$ is $\left[ -5,7 \right]$. Therefore, $g\left( x \right)=\left| 2x+5 \right|$ has to be in the range of $\left[ -5,7 \right]$.
This means $-5\le \left| 2x+5 \right|\le 7$. We break it in two parts of $-5\le \left| 2x+5 \right|$ and $\left| 2x+5 \right|\le 7$.
From $-5\le \left| 2x+5 \right|$, we get $x\in \mathbb{R}$ as the modulus value is always positive.
From $\left| 2x+5 \right|\le 7$, we get
$\begin{aligned} & \left| 2x+5 \right|\le 7 \\\ & \Rightarrow -7\le 2x+5\le 7 \\\ & \Rightarrow -12\le 2x\le 2 \\\ & \Rightarrow -6\le x\le 1 \\\ \end{aligned}$
Therefore, $x\in \left[ -6,1 \right]$.
The intersection of $x\in \left[ -6,1 \right]$ and $x\in \mathbb{R}$ gives $x\in \left[ -6,1 \right]$. Therefore, the correct option is C.

Note:
We need to remember that $x\in \left[ -6,1 \right]$ is solely responsible for the expression to be defined. We also considered that complex values as the root are not allowed. In the first case the domain becomes the whole real line.