Let $f(x)=e^x+x$. If $g(x)$ is the inverse function of $f(x)$, then $g^{\prime }(1)$ is:

Solution

1. Since $g$ is the inverse of $f$, $g^{\prime }(y)=\frac{1}{f^{\prime }(x)}$ where $f(x)=y$.
2. We want $g^{\prime }(1)$, so set $$\begin{array}{rl}f(x)& =e^x+x\\ & =1\end{array}$$.
3. By inspection, $x=0$ satisfies $e^0+0=1$.
4. Calculate $f^{\prime }(x)=e^x+1$.
5. At $x=0$, $$\begin{array}{rl}{f}^{\prime }(0)& =e^0+1\\ & =2\end{array}$$, so $g^{\prime }(1)=1/2$. Hence the answer is (A).