Answer
- Figure out the formula of the inverse of $f(x)=mx+b$ with $m\ne0$.
- Looking at the formula, we can deduce the shape of the graph, as well as its slope and its $y$-intercept.
Work Step by Step
$$y=f(x)=mx+b\hspace{1cm}m\ne0$$
(a) To find its inverse:
1) Solve for $x$ in terms of $y$:
$$y=mx+b$$ $$mx=y-b$$ $$x=\frac{y-b}{m}$$
2) Interchange $x$ and $y$:
$$y=\frac{x-b}{m}$$
Therefore, $$f^{-1}(x)=\frac{x-b}{m}$$
Here, we can rewrite the formula of the inverse: $$f^{-1}(x)=\frac{1}{m}x-\frac{b}{m}$$
Looking at the formula, we can conclude that the graph of the inverse is a line, with the slope $1/m$. The $y$-intercept is $-b/m$, as expected by the exercise.
