Answer
a) $f(0)=1$
b) $f^{-1}(1)=0$
Work Step by Step
Substituting $x=
0
$ in the given function, $
f(x)=2^x
,$ results to
\begin{align*}\require{cancel}
f(0)&=2^0
\\&=
1
.\end{align*}
Hence, $
f(0)=1
$.
The point $(x,f(x))=
(0,1)
$ 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 $
(1,0)
$, is on the inverse $f^{-1}(x).$ Hence, $
f^{-1}(1)=0
$.
