Given $g: \Bbb R \to \Bbb R^+$ where $g(x) = 3^x$, define inverse function $g^{-1}$

Question

Given $g: \Bbb R \to \Bbb R^+$ where $g(x) = 3^x$, define fully the inverse function $g^{-1}$ and state the value of $g^{-1}(1)$.

This is what I have tried:

$$\begin{align} g(x) &= 3^x \\ 3^x &= y \\ x \log_3 3 &= \log_3 y \\ x &= \log_3 y \\ g^{-1}(1) &= \log_3 1 \end{align}$$

Why is this wrong?

Answer 1

It's not untrue, but there are two reasons it might be marked wrong in a test.

1. You arguably didn't answer the first part of the question: The inverse of $g(x)$ is $g: \mathbb{R}^+ \rightarrow \mathbb{R}$, $g^{-1}(x) = \log_3 x$.
2. You didn't simplify your answer to $0$.

Answer 2

This seems to be correct to me. The only thing that I think could have puzzled your professor is that $g^{-1}(1)=0$, but your answer isn't wrong as it too equals zero.