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The graph of the exponential function $f(x)$ is shown below.  Find $f(x).$

 

The function passes through (1,3) and (2,1/3).

 Feb 17, 2024
 #1
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The general form of an exponential function is usually \(y=a\cdot{b}^{x}\). (Notice I didn't set it as \(y=a\cdot{b}^{x}+k\).

We plug our values in,

 \(\begin{cases} 3=a\cdot{b}^{1} \\ \frac{1}{3}=a\cdot{b}^{2} \end{cases} \)

\(\begin{cases} ab=3 \\ a{b}^{2}=\frac{1}{3} \end{cases} \)

Subbing 1 into 2, 

\(3b=\frac{1}{3}\)

\(b=\frac{1}{9}\)

Subbing b back into equation 1, we get

\(a=27\)

So our exponential fuction is \(y=27\cdot{(\frac{1}{9})}^{x}\).

But note, there are infinitely many solutions to this equation, this is just one of the solutions, in the standard form without translations.

 Feb 17, 2024

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