Question

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

• A. $-\sqrt{x}-1,x\ge 0$
• B. $\frac{1}{(x+1{)}^2},x>-1$
• C. $\sqrt{x+1},x\ge -1$
• D. $\sqrt{x}-1,x\ge 0$

Answer 1

The correct option is D $\sqrt{x}-1,x\ge 0$

Reflection of a curve about $y=x$ line is same as the inverse of the same function.
Let $y=f\left(x\right)=(x+1{)}^2$ for $x\ge -1$
$\pm \sqrt{y}=x+1,x\ge -1$
$\Rightarrow \sqrt{y}=x+1\Rightarrow y\ge 0,x+1\ge 0\Rightarrow x=\sqrt{y}-1\Rightarrow f^{-1}\left(y\right)=\sqrt{y}-1\Rightarrow f^{-1}\left(x\right)=\sqrt{x}-1\Rightarrow x\ge 0$