Answer
See below for detailed answers.
Work Step by Step
$$f(x)=5-4x$$
a) To find $f^{-1}(x)$, we substitute $x$ with $f^{-1}x$ and $f(x)$ with $x$: $$x=5-4f^{-1}(x)$$
Then solve for $f^{-1}(x)$: $$4f^{-1}(x)=5-x$$ $$f^{-1}(x)=\frac{5-x}{4}$$
b) The graphs of $f(x)$ and $f^{-1}(x)$ are enclosed below.
c) - Find $df/dx$: $$\frac{df}{dx}=\frac{d}{dx}\Big(5-4x\Big)=-4$$
At $x=1/2$, $df/dx=-4$
- Find $df^{-1}/dx$:
$$\frac{df^{-1}}{dx}=\frac{d}{dx}\Big(\frac{5-x}{4}\Big)=\frac{1}{4}(0-1)=-\frac{1}{4}$$
At $x=f(1/2)=5-4\times(1/2)=3$, $$df^{-1}/dx=-\frac{1}{4}$$
It can be seen that at these points, $df^{-1}/dx=1/(df/dx)$
