Sheet № 97 · Higher only · AQA · Edexcel · OCR
Inverse Functions –
Inverse functions reverse the effect of the original function. If f takes 3 to 11, then the inverse function f⁻¹ takes 11 back to 3. This is a Higher-tier topic that appears on AQA, Edexcel and OCR papers, typically worth three to four marks.
§Key definitions
Question:
f(x) = 3x + 7. Find f⁻¹(x).
Answer:
f⁻¹(x) = (x − 7)/3
Q1 (Higher):
f(x) = 5x − 4. Find f⁻¹(x).
Q2 (Higher):
f(x) = (x + 3)/2. Find f⁻¹(x) and evaluate f⁻¹(7).
Q3 (Higher):
g(x) = x² − 1 for x ≥ 0. Find g⁻¹(x).
§Formulas to memorise
If f(a) = b, then f⁻¹(b) = a
To find f⁻¹(x): write y = f(x), swap x and y, rearrange for y
Write — y = f(x) — replace f(x) with y.
Swap — x and y in the equation.
Rearrange — the new equation to make y the subject.
Write — f⁻¹(x) = the expression you found for y.
Worked example
f(x) = 3x + 7. Find f⁻¹(x).
Working:
⚠ Common mistakes
- ✗Confusing f⁻¹(x) with 1/f(x). The notation f⁻¹(x) means the inverse function, not the reciprocal. f⁻¹(x) undoes f, whereas 1/f(x) divides 1 by the output.
- ✗Forgetting to swap x and y. Simply rearranging y = 2x + 3 to get x = (y − 3)/2 is not enough. You must swap the variables first, then rearrange for y.
- ✗Making rearrangement errors with fractions. When the function involves a fraction, multiply both sides first to clear the denominator before isolating y.
✦ Exam tips
- →Always verify your inverse by checking that f(f⁻¹(x)) = x. Pick a simple number and confirm that applying f then f⁻¹ gets you back to where you started.
- →If a question asks you to sketch f⁻¹(x), reflect the graph of f(x) in the line y = x. Key points swap their coordinates.
- →Show each algebraic step clearly — write the swap step explicitly so the examiner can see your method.