Question

Let $f\left(x\right)=(x+1{)}^2$ for $x\ge -1$. If $g\left(x\right)$ is a function whose graph is the reflection of the graph of$f\left(x\right)$in the line $y=x$, then $g\left(x\right)$ is equal to

• A. $\left[\frac{1}{(x+1{)}^2}\right]x>-1$
• B. $-\sqrt{x}-1$
• C. $\sqrt{x}+1$
• D. $\sqrt{x}-1$

Answer 1

The correct option is D

$\sqrt{x}-1$

Explanation for Correct Answer:

Given that: $f\left(x\right)=(x+1{)}^2$ for all $x\ge -1$

$g\left(x\right)$ is a reflection of $f\left(x\right)$in $y=x$.

This means $g\left(x\right)=f^{-1}\left(x\right)$.

Calculating $f^{-1}\left(x\right)$:

Let

$\begin{array}{l}y=(x+1{)}^2\\ ⟹\pm \sqrt{y}=(x+1)\\ x=\sqrt{y}-1\\ \therefore f^{-1}\left(y\right)=\sqrt{y}-1\\ \therefore f^{-1}\left(x\right)=\sqrt{x}-1\end{array}$

We can say,

$g\left(x\right)=f^{-1}\left(x\right)=\sqrt{x}-1$

Hence, option (D) is the correct answer