+1768 The graph of the exponential function $f(x)$ is shown below. Find $f(x).$
The function passes through (1,3) and (2,1/3).
+399 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.
