Inverse Function

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Inverse Function: meaning, method and solved example (Algebra 1) – Maths PDF

In short: An inverse function reverses a given function: whenever f sends an input x to an output y, the inverse takes y and returns exactly that x.

Algebra 1Functions and GraphsF-BF.B.4a

What is Inverse Function?

An inverse function plays a rule backward. When a parking garage turns hours into dollars, the backward rule turns dollars into hours, so the roles of input and output trade places.

For a function \(f\), the backward rule \(f^{-1}\) obeys \(f^{-1}(f(x)) = x\) and \(f(f^{-1}(x)) = x\). Plotted, it mirrors the graph of \(f\) over the diagonal \(y = x\). Only a one-to-one function, with each output produced by a single input, owns an inverse that is again a function.

How to spot it

  • A formula converts one quantity into another, and you must now convert in the opposite direction; that is an inverse function request.
  • You are given an output and asked which input produced it, such as the hours that led to a $34 fee.
  • The notation \(f^{-1}(x)\) appears, or the problem says to swap x and y.
  • Two graphs look like mirror images of each other across the diagonal line y = x.

The method

  1. Write the function as an equation \(y = f(x)\), with the output variable alone on the left.
  2. Swap \(x\) and \(y\) everywhere in the equation.
  3. Solve the new equation for \(y\), undoing operations in reverse order.
  4. Name the result \(f^{-1}(x)\) and test it: \(f(f^{-1}(x))\) must simplify to \(x\).
(2, 5)(5, 2)f: y = 2x + 1inverse: y = (x – 1)/2mirror: y = x
The two lines are reflections of each other across the dashed green line y = x.

Inverse Function: solved example

Problem

A downtown lot bills \(f(h) = 3.5h + 6\) dollars for \(h\) hours. Build the reverse rule, then use it to learn how long a car stayed when the receipt says $34.

Solution

Write \(y = 3.5h + 6\), swap the variables: \(h = 3.5y + 6\). Subtract 6 and divide by 3.5: \(y = \frac{h – 6}{3.5}\), so \(f^{-1}(h) = \frac{h – 6}{3.5}\), where the input is now dollars.

Check: \(f(8) = 3.5(8) + 6 = 34\). Then \(f^{-1}(34) = \frac{28}{3.5} = 8\), so the driver parked 8 hours.

Try it

Question 1

Find the inverse of \(f(x) = 4x – 9\).

Show the answer

Swap: \(x = 4y – 9\), so \(y = \frac{x + 9}{4}\) and \(f^{-1}(x) = \frac{x + 9}{4}\).

Question 2

If \(g(3) = 11\), what is \(g^{-1}(11)\)?

Show the answer

The inverse reverses the pairing, so \(g^{-1}(11) = 3\).

Inverse Function vs. Reciprocal

The exponent in \(f^{-1}(x)\) is a name tag for the inverse, not a power. It does not mean \(\frac{1}{f(x)}\). For \(f(x) = 2x + 1\), the inverse is \(\frac{x – 1}{2}\), while the reciprocal is \(\frac{1}{2x + 1}\); at \(x = 5\) these give 2 and \(\frac{1}{11}\).

Reciprocal

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