Answer
See below for detailed answers.
Work Step by Step
$$f(x)=2x+3$$
a) To find $f^{-1}(x)$, we substitute $x$ with $f^{-1}x$ and $f(x)$ with $x$: $$x=2f^{-1}(x)+3$$
Then solve for $f^{-1}(x)$: $$f^{-1}(x)=\frac{x-3}{2}$$
b) The graphs of $f(x)$ and $f^{-1}(x)$ are enclosed below.
c) - Evaluate $df/dx$ at $x=-1$: $$\frac{df}{dx}=2\times1+0=2$$
At $x=-1$, $df/dx=2$
- Evaluate $df^{-1}/dx$ at $x=f(-1)=2(-1)+3=1$:
$$\frac{df^{-1}}{dx}=\frac{1}{2}(1-0)=\frac{1}{2}$$
At $x=1$, $df^{-1}/dx=1/2$
It can be seen that at these points, $df^{-1}/dx=1/(df/dx)$
