Knowledge Organiser: Inverse Functions
This topic summary covers Knowledge Organiser: Inverse Functions within Inverse Functions for GCSE Mathematics. Revise Inverse Functions in Algebra for GCSE Mathematics with 8 exam-style questions and 4 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 7 of 7 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Topic position
Section 7 of 7
Practice
8 questions
Recall
4 flashcards
Knowledge Organiser: Inverse Functions
Key Terms
- Inverse function f⁻¹(x): Reverses what f(x) does — maps outputs back to inputs
- One-to-one function: Each input gives a unique output (invertible)
- Reflection: The graph of f⁻¹(x) is the reflection of f(x) in the line y = x
- Self-inverse: A function that is its own inverse (e.g. f(x) = 1/x)
- f⁻¹ notation: The superscript −1 means inverse, NOT "to the power of −1"
Must-Know Facts
- f⁻¹(x) is NOT the same as 1/f(x) (inverse ≠ reciprocal)
- Method: write y = f(x), swap x and y, rearrange for y
- The graph of f⁻¹(x) is a reflection of f(x) in the line y = x
- f(f⁻¹(x)) = x and f⁻¹(f(x)) = x — they undo each other
- Always verify your inverse by checking one value works
Key Methods
- Step 1: Write y = f(x)
- Step 2: Swap x and y (write x = f(y))
- Step 3: Rearrange to make y the subject
- Step 4: Write f⁻¹(x) = your rearranged expression
- Verify: substitute a value and check f(f⁻¹(value)) = value
Common Mistakes
- Confusing inverse with reciprocal: f⁻¹(x) is NOT 1/f(x) — the −1 superscript means inverse function, not a power of −1
- Forgetting to swap x and y: The key step is writing x = f(y) — skipping the swap means you rearrange the original function, not its inverse
- Errors rearranging: Take extra care with fractions and square roots when making y the subject — check your rearrangement by substituting a test value
- Applying f⁻¹ in the wrong order for composition: ff⁻¹(x) = x, but be careful that f⁻¹f(x) also = x — both compositions return x
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Practice Questions for Inverse Functions
What does f⁻¹(x) represent?
Explain why the function f(x) = x² (for all real x) does not have an inverse function over its full domain.
Quick Recall Flashcards
8 questions on Inverse Functions — practise free
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