Answer
$ a.\quad f$ is one-to-one
$b.\quad f^{-1}(x)=\displaystyle \frac{4}{x}$
Work Step by Step
$ a.\quad$
The function is rational, with a graph that passes the horizontal line test (see below)
(It is impossible to draw a horizontal line that intersects a function's graph more than once.)
It is one-to-one and has an inverse.
$ b.\quad$
To find a formula for the inverse,
1. Replace $f(x)$ with $y.$
$y=\displaystyle \frac{4}{x}$
2. Interchange $x$ and $y$. (This gives the inverse function.)
$x=\displaystyle \frac{4}{y}$
3. Solve for $y.$
... multiply with $\displaystyle \frac{y}{x}$
$y=\displaystyle \frac{4}{x}$
4. Replace $y$ with $f^{-1}(x)$ . (This is inverse function notation.)
$f^{-1}(x)=\displaystyle \frac{4}{x}$
