Answer
a) Finding the inverse function
$f^{-1}(x) = \frac{1}{4}x$
b) Verifying that it is the correct equation
$f(f^{-1}(x)) = x$
$f(\frac{x}{4}) = 4\frac{x}{4}$
$= \frac{4x}{4}$
$= x$
$f^{-1}(f(x)) = x$
$f^{-1}(4x) = \frac{1}{4}(4x)$
$= \frac{4x}{4}$
$= x$
Therefore, it is the correct inverse equation.
Work Step by Step
a) Find the inverse
Let $f(x) = y$
$y = 4x$
$x = 4y$
$\frac{x}{4} = y$
$y = \frac{1}{4}x$
$f^{-1}(x) = \frac{1}{4}x$
b) Verifying that it is the correct equation
$f(f^{-1}(x)) = x$
$f(\frac{x}{4}) = 4\frac{x}{4}$
$= \frac{4x}{4}$
$= x$
$f^{-1}(f(x)) = x$
$f^{-1}(4x) = \frac{1}{4}(4x)$
$= \frac{4x}{4}$
$= x$
Therefore, it is the correct inverse equation.