Inverse Functions for IGCSE Maths
Finding and using the inverse function f inverse of x. This subtopic is part of Functions in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding inverse functions is esse
What You Need to Know
Finding and using the inverse function f inverse of x. This subtopic is part of Functions in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding inverse functions is essential for achieving a strong grade in your IGCSE Maths exam.
Understanding Inverse Functions
The inverse function f⁻¹(x) reverses what f does: if f(a) = b, then f⁻¹(b) = a. To find f⁻¹(x), replace f(x) with y, swap x and y, then make y the subject — this new y is f⁻¹(x). The domain of f⁻¹ is the range of f. In IGCSE Extended, inverse functions appear in Paper 4 and are sometimes combined with composite functions in multi-part questions.
Step-by-Step Method
- 1
Write y = f(x)
Replace f(x) with y to make the notation easier to work with.
- 2
Swap x and y
Write the equation with x and y exchanged. This is the key step that creates the inverse.
- 3
Make y the subject
Rearrange to isolate y using the standard rearranging techniques (inverse operations, factorisation).
- 4
Write f⁻¹(x) = ...
Replace y with f⁻¹(x) to complete the answer.
- 5
Verify with a test value
If f(3) = 7, then f⁻¹(7) should equal 3. Check this with your inverse function.
Worked Example
Question
f(x) = (2x − 3)/(x + 1). Find f⁻¹(x).
Solution
Step 1: Let y = (2x − 3)/(x + 1) Step 2: Swap x and y: x = (2y − 3)/(y + 1) Step 3: Make y the subject. Multiply both sides by (y + 1): x(y + 1) = 2y − 3 xy + x = 2y − 3 xy − 2y = −3 − x y(x − 2) = −3 − x y = (−3 − x)/(x − 2) or equivalently (x + 3)/(2 − x) Step 4: f⁻¹(x) = (x + 3)/(2 − x) Verify: f(3) = (6−3)/(3+1) = 3/4. Check f⁻¹(3/4) = (3/4+3)/(2−3/4) = (15/4)/(5/4) = 3 ✓
Exam Tips for Inverse Functions
- The swap-x-and-y step is the essential one — do it explicitly, not in your head.
- f⁻¹(x) means the inverse function, NOT 1/f(x). These are completely different things.
- After finding f⁻¹, verify with one numerical value: compute f(a) and check that f⁻¹(f(a)) = a.
- When f(x) is a fraction with x in both numerator and denominator, expect to factorise in the rearranging step.
Practice Questions
Q1: f(x) = 4x − 7. Find f⁻¹(x) and evaluate f⁻¹(5).
Show hint
y=4x−7 → x=4y−7 → y=(x+7)/4. So f⁻¹(5) = 12/4 = 3.
Q2: g(x) = x² + 1 for x ≥ 0. Find g⁻¹(x) and state its domain.
Show hint
y=x²+1 → x=y²+1 → y=√(x−1). Domain of g⁻¹ is x ≥ 1 (the range of g).
Q3: Solve ff⁻¹(x) = 7.
Show hint
ff⁻¹ is the identity function, so ff⁻¹(x) = x. Equation becomes x = 7.
Frequently Asked Questions
What is inverse functions in IGCSE Maths?
Finding and using the inverse function f inverse of x.
Is inverse functions in the Core or Extended syllabus?
Inverse Functions is part of the Extended only syllabus for IGCSE Mathematics 0580.
How do I revise inverse functions effectively?
Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on inverse functions rather than trying to cover everything at once.
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