Answer
a) Finding the inverse function
$f^{-1}(x) = \frac{1}{2}x$
b) Verifying that it is the correct equation
$f(f^{-1}(x)) = x$
$f(\frac{x}{2}) = 2\frac{x}{2}$
$= \frac{2x}{2}$
$= x$
$f^{-1}(f(x)) = x$
$f^{-1}(2x) = \frac{1}{2}(2x)$
$= \frac{2x}{2}$
$= x$
Therefore, it is the correct inverse equation.
Work Step by Step
a) Find the inverse
Let $f(x) = y$
$y = 2x$
$x = 2y$
$\frac{x}{2} = y$
$y = \frac{1}{2}x$
$f^{-1}(x) = \frac{1}{2}x$
b) Verifying that it is the correct equation
$f(f^{-1}(x)) = x$
$f(\frac{x}{2}) = 2\frac{x}{2}$
$= \frac{2x}{2}$
$= x$
$f^{-1}(f(x)) = x$
$f^{-1}(2x) = \frac{1}{2}(2x)$
$= \frac{2x}{2}$
$= x$
Therefore, it is the correct inverse equation.