How to Calculate Inverse Functions (IGCSE Mathematics)
Knowing how to calculate inverse functions is a core Edexcel IGCSE Maths skill: an inverse, written f⁻¹(x), reverses what a function does, sending each output back to its input. This page shows the reliable method (swap x and y, then make y the subject), how to handle the domain and range, and how to check your inverse by composing it with the original. Worked examples and the auto-marked practice rooms below give instant feedback.
An inverse function reverses what the original function does. If \(f\) maps an input \(x\) to an output \(y\),
then \(f^{-1}\) maps \(y\) back to \(x\). In other words, applying \(f\) then \(f^{-1}\) (or vice versa)
returns the original value.
Notation: the inverse of \(f(x)\) is written \(f^{-1}(x)\).
Note that \(f^{-1}(x)\) does not mean \(\dfrac{1}{f(x)}\).
How to Find an Inverse Function
Write the function as \(y = f(x)\).
Swap \(x\) and \(y\) to get \(x = f(y)\).
Rearrange to make \(y\) the subject.
Rename \(y\) as \(f^{-1}(x)\).
Check by composition: \(f\!\left(f^{-1}(x)\right) = x\).
Core Ideas
Inverse undoes the function
Applying \(f\) then \(f^{-1}\) returns \(x\): \(f\!\left(f^{-1}(x)\right) = x\).
Swap, then solve
Start with \(y = f(x)\), swap \(x\) and \(y\), then rearrange for \(y\).
Domains and ranges swap
The domain of \(f\) becomes the range of \(f^{-1}\), and vice versa.
Worked Examples
💡 Example 1: Linear (Room 1)
Find \(f^{-1}(x)\) where \(f(x) = 4x + 7\).
Write \(y = f(x)\)\(y = 4x + 7\)
Swap \(x\) and \(y\)\(x = 4y + 7\)
Subtract 7 from both sides\(x - 7 = 4y\)
Divide both sides by 4\(y = \dfrac{x-7}{4}\)
Rename \(y\) as \(f^{-1}(x)\)\(f^{-1}(x) = \dfrac{x-7}{4}\)
💡 Example 2: Reciprocal (Room 2)
Find \(f^{-1}(x)\) where \(f(x) = 5 + \dfrac{6}{x}\).
Write \(y = f(x)\)\(y = 5 + \dfrac{6}{x}\)
Swap \(x\) and \(y\)\(x = 5 + \dfrac{6}{y}\)
Subtract 5 from both sides\(x - 5 = \dfrac{6}{y}\)
Multiply by \(y\), then divide by \((x-5)\)\(y = \dfrac{6}{x-5}\)
Rename \(y\) as \(f^{-1}(x)\)\(f^{-1}(x) = \dfrac{6}{x-5}\)
💡 Example 3: Rational function (Room 3)
Find \(f^{-1}(x)\) where \(f(x) = \dfrac{3x+2}{x-1}\).
Write \(y = f(x)\)\(y = \dfrac{3x+2}{x-1}\)
Swap \(x\) and \(y\)\(x = \dfrac{3y+2}{y-1}\)
Multiply both sides by \((y-1)\)\(x(y-1) = 3y + 2\)
Collect the \(y\) terms\(y(x-3) = x + 2\)
Divide by \((x-3)\) and rename\(f^{-1}(x) = \dfrac{x+2}{x-3}\)
💡 Example 4: Square root (Room 4)
Find \(f^{-1}(x)\) where \(f(x) = \sqrt{x+5}\).
Write \(y = f(x)\)\(y = \sqrt{x+5}\)
Swap \(x\) and \(y\)\(x = \sqrt{y+5}\)
Square both sides\(x^2 = y + 5\)
Subtract 5 to make \(y\) the subject\(y = x^2 - 5\)
Rename (domain \(x \geq 0\), the range of \(f\))\(f^{-1}(x) = x^2 - 5\)
🔑 Key Points
Swap \(x\) and \(y\), then rearrange to find the inverse.
Check: \(f\!\left(f^{-1}(x)\right) = x\).
The graph of \(f^{-1}\) is a reflection of \(f\) in the line \(y = x\).
Domains and ranges swap between \(f\) and \(f^{-1}\).
⚠️ Common Pitfalls
Rearranging without swapping \(x\) and \(y\) first.
Confusing \(f^{-1}(x)\) with \(\dfrac{1}{f(x)}\): they are different things.
Forgetting domain restrictions when inverting quadratics (two branches exist).
Rooms 1-4: find \(f^{-1}(x)\) and enter the expression only, not f⁻¹(x) =
(any correct equivalent form is accepted), e.g. (x-3)/2, 6/(x+1),
x^2-4, sqrt(x)+3.
Room 5 evaluates the inverse at a value and Room 6 solves \(f(x)=f^{-1}(x)\): type a number
(two, where a card gives two boxes). Room 7 mixes every type.
How to calculate inverse functions: Step by step video guide
This YouTube video provides a comprehensive explanation of inverse functions, along with visual examples and step-by-step solutions. It covers key concepts related to inverse functions and offers additional insights to enhance your understanding. Take the time to watch the video, and you’ll gain a solid foundation in solving inverse functions.
