Answer
(a) $$f^{-1}(x)=\frac{x}{m}$$
(b) The graph of the inverse of $y=f(x)=mx$ is a line through the origin with a non-zero slope $\frac{1}{m}$
Work Step by Step
$$y=f(x)=mx\hspace{1cm}m\ne0$$
(a) To find its inverse:
1) Solve for $x$ in terms of $y$:
$$y=mx$$ $$x=\frac{y}{m}$$
2) Interchange $x$ and $y$:
$$y=\frac{x}{m}$$
Therefore, $$f^{-1}(x)=\frac{x}{m}$$
(b) As told in the exercise, $f(x)=mx$ in the graph is a line through the origin with a non-zero slope $m$.
We found its inverse, which is $f^{-1}(x)=\frac{x}{m}$
If we replace $x=0$, we find that $f^{-1}(0)=\frac{0}{m}=0$. So the graph of the inverse also goes through the origin.
Looking at the formula, we figure that the graph of the inverse is also a line, too, with a non-zero slope $\frac{1}{m}$
Therefore, we can conclude that the graph of the inverse of $y=f(x)=mx$ is a line through the origin with a non-zero slope $\frac{1}{m}$
