Inverse functions
An inverse reverses the input-output relationship on a one-to-one domain.
Write y = f(x), solve for x in terms of y, then exchange the variable names. The domain of the inverse is the range of the original function. The notation f⁻¹ means inverse function here, not reciprocal.
If two allowed inputs have the same output, there cannot be a single inverse function on that domain. A quadratic needs a restricted branch. Use the domain restriction to choose the sign of the square root.
Worked example
Find the inverse of f(x) = (x − 1)² for x ≥ 1.
- Take the nonnegative root because x − 1 ≥ 0.
- √y = x − 1, so x = 1 + √y.
- The original range is y ≥ 0.
Answer: f⁻¹(x) = 1 + √x for x ≥ 0
Revise first: Domain and range.
Practise inverse functions
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 8365 · core · 2.5: Inverse functions
- 7M20 · core · A2.3: Inverse functions
- 8365 · core · 2.5: Inverting a linear function
- 9FM0 · core · Assumed A-level Mathematics knowledge: Inverting a linear function
- 7367 · core · Assumed A-level Mathematics knowledge: Inverting a linear function
- H245 · core · Assumed A-level Mathematics knowledge: Inverting a linear function
- H645 · core · Assumed A-level Mathematics knowledge: Inverting a linear function
- 7M20 · core · A2.3: Inverting a linear function
- 8365 · core · 2.5: Inverting a restricted quadratic
- 9FM0 · core · Assumed A-level Mathematics knowledge: Inverting a restricted quadratic
- 7367 · core · Assumed A-level Mathematics knowledge: Inverting a restricted quadratic
- H245 · core · Assumed A-level Mathematics knowledge: Inverting a restricted quadratic
- H645 · core · Assumed A-level Mathematics knowledge: Inverting a restricted quadratic
- 7M20 · core · A2.3: Inverting a restricted quadratic
Next practice: Composite functions.