# Chapter 4, Exponential and Logarithmic Functions - Section 4.1 - Exponential Functions - 4.1 Exercises: 23

$\color{blue}{f(x)=\left(\dfrac{1}{4}\right)^x}$

#### Work Step by Step

The graph of the function $f(x)=a^x$ contains the point $(2,\frac{1}{16})$. This means that when $x=2$, $y = \frac{1}{16}$. Substitute $x$ and $y$ into $f(x) =a^x$ to obtain: $\begin{array}{ccc} \\&f(x) &= &a^x \\&f(2) &= &a^{2} \\&\dfrac{1}{16} &= &a^{2}\end{array}$ Note that $\dfrac{1}{16} = \left(\dfrac{1}{4}\right)^2$. Thus, the expression above is equivalent to: $\left(\dfrac{1}{4}\right)^2=a^2$ Use the rule "$a^m=b^m \longrightarrow a=b$" to obtain: $\dfrac{1}{4} = a$ With $a=\frac{1}{4}$, the function whose graph is given is $\color{blue}{f(x)=\left(\dfrac{1}{4}\right)^x}$.

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