Answer
$\color{blue}{f(x)=5^x}$.
Work Step by Step
The graph of the function $f(x)=a^x$ contains the point $(-1,\frac{1}{5})$.
This means that when $x=-1$, $f(-1) = \frac{1}{5}$.
Substitute $x$ and $y$ into $f(x) =a^x$ to obtain:
$\begin{array}{ccc}
\\&f(x) &= &a^x
\\&f(-1) &= &a^{-1}
\\&\dfrac{1}{5} &= &a^{-1}\end{array}$
Use the rule $a^{-m}=\dfrac{1}{a^m}$ to obtain:
$\dfrac{1}{5} = \dfrac{1}{a}$
Cross-multiply to obtain:
$1(a) = 1(5)
\\a=5$
With $a=5$, the function whose graph is given is $\color{blue}{f(x)=5^x}$.
