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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

Whiteboard maths

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