Answer
a) $f(-2)=\dfrac{1}{4}$
b) $f^{-1}\left(\dfrac{1}{4}\right)=-2$
Work Step by Step
Substituting $x=
-2
$ in the given function, $
f(x)=2^x
,$ results to
\begin{align*}\require{cancel}
f(-2)&=2^{-2}
\\\\&=
\dfrac{1}{2^2}
\\\\&=
\dfrac{1}{4}
.\end{align*}
Hence, $
f(-2)=\dfrac{1}{4}
$.
The point $(x,f(x))=
\left(-2,\dfrac{1}{4}\right)
$ is a point on the given function. Since $f(x)$ is given to be one-to-one, then by interchanging the $x$ and $y$ coordinates, the point $
\left(\dfrac{1}{4},-2\right)
$, is on the inverse $f^{-1}(x).$ Hence, $
f^{-1}\left(\dfrac{1}{4}\right)=-2
$.
