Inverse Functions (Economics)
Inverse Functions
An inverse function “undoes” the work of a function. We typically denote inverse of the function $f(x)$ by $f^{-1}(x)$.
For example, if a function $f(x)$ added $2$ to $x$ and then squared the result, the corresponding inverse function $f^{-1}(x)$ would take the square root of $x$ and then subtract $2$: \[f(x)=(x+2)^2\] \[f^{-1}(x)=\sqrt{x}-2\]
Because the inverse of a function does the opposite to the function, if we apply a function and then apply its inverse, we should get back the original value: \[f^{-1}(f(x))=x\] For example, consider the function $g(x)=10x$. The inverse of this function is $g^{-1}(x)=\frac{x}{10}$. Applying $g$ to $x$ and then $g^{-1}$ gives: \[\dfrac{10x}{10}=x\]
Note: The $-1$ in $f^{-1}(x)$ is not a power.
Finding the Inverse Function
For example, suppose we want to find the inverse of the function $f(x)=x^2-1$. We can do this using the following three steps:
1) Replace $f(x)$ with $y$.
2) Swap the independent variable $x$ with the dependent variable $y$. This gives $x=y^2-1$.
3) Rearrange the function to make dependent variable $y$ the subject. This gives $y=\sqrt{x+1}$.
4) Finally, replace $y$ with $f^{-1}(x)$. The inverse of $f(x)=40x-9$ is therefore $f^{-1}(x)=\sqrt{x+1}$
Graphing the Inverse Function
We can draw the graph of the inverse of a function by reflecting the graph of the function in the diagonal line $y=x$. The graph below shows the curves $f(x) = x^2-1$ in red, and $f^{-1}(x) = \sqrt{x+1}$ in blue. The dotted line is $y=x$.
Worked Example
Question
Find the inverse function of $f(x) =4-\dfrac{5}{x}$. Plot the function and its inverse in the same graph.
Solution
Following the above steps, we have:
1) $y=4-\dfrac{5}{x}$
2) $x=4-\dfrac{5}{y}$
3) \begin{align} x&=4-\dfrac{5}{y}\\ \Rightarrow x&+\dfrac{5}{y}=4\\ \Rightarrow \dfrac{5}{y}&=4-x\\ \Rightarrow \dfrac{y}{5}&=\dfrac{1}{4-x}\\ \Rightarrow y&=\dfrac{5}{4-x}\\ \end{align}
4) $f^{-1}(x) = -\dfrac{5}{x-4}$
This graph shows the lines $f(x) = 4 - \dfrac{5}{x}$ in red, and $f^{-1}(x) = \dfrac{5}{x-4}$ in blue. The dotted line is $y=x$.
Video Example
Prof. Robin Johnson finds the inverse of the function $f(x) = 2 - \dfrac{1}{x}$.
Workbook
This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.
External Resources
- Inverse functions workbook at mathcentre.
