Inverse Functions¶
Given a function, you can inverse it as so that the input of the inverse function is the output of the original function, and the output of the inverse function is the input of the original function.
Inverse Functions are denoted by \(name^{-1}(inputs...) = output\).
Example¶
Given the function \(f(x) = 3x + 2\), you can work out the inverse with the following procedure:
- Change the function as equal to \(y\)
- Rearrange the function to make
inputthe subject - Swap \(x\) out for
name(input)and \(y\) withinput
\[
\begin{align}
f(x) & = 3x + 2 \\
y & = 3x + 2 \\
y - 2 & = 3x \\
\frac{y - 2}{3} & = x \\
f^{-1}(x) & = \frac{x - 2}{3}
\end{align}
\]
Lets test the function.
\[
\begin{align}
f(x) & = 3x + 2 \\
f(3) & = 3(3) + 2 \\
& = 11 \\ \\
f^{-1}(x) & = \frac{x - 2}{3} \\
f^{-1}(11) & = \frac{11 - 2}{3} \\
& = \frac{9}{3} \\
& = 3
\end{align}
\]
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