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

The graph of $f\left(x\right)$ has the equation $y=(x+1{)}^2, x\ge -1$

The reflection of the graph of $f\left(x\right)$ is obtained by 

interchanging $x,y$ So, the graph of $g\left(x\right)$ has the equation $x=(y+1{)}^2, y\ge -1$

$\therefore y=\sqrt{x}-1$ because $y\ge -1$ 

$\therefore \varphi \left(x\right)=\sqrt{x}-1, x\ge 0.$