Maths Extension 1 Year 11 - Module 1 - Lesson 6
Domain Restriction for Inverses
Name
Date
Class
1. Lesson focus
Domain restriction for inverse functions. A quadratic is one-to-one on a restricted half-domain; choose the matching sign when solving its inverse.
Success criteria
- Apply the lesson method accurately in a routine example.
- Show each algebraic or domain step and check the result.
- Explain why the method and restrictions fit the question.
2. Worked practice
2. For f(x)=(x−3)²+1 restricted to x≥3, find its inverse.
3. For f(x)=x² restricted to x≤0, find its inverse.
3. Challenge
Find the inverse of f(x)=1/(x−2) restricted to x>2.
4. Multiple choice
For f(x)=x² on x≥0, the range (and inverse domain) is which?
A. [0,∞)
B. (-∞,∞)
C. (-∞,0]
D. [1,∞)
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Restrict f(x)=x² to x≥0. Find f⁻¹.
y=x² and x≥0, so x=√y; f⁻¹(x)=√x with domain x≥0.
2. For f(x)=(x−3)²+1 restricted to x≥3, find its inverse.
y=(x−3)²+1; x−3=√(y−1), so f⁻¹(x)=3+√(x−1), x≥1.
3. For f(x)=x² restricted to x≤0, find its inverse.
x=−√y; therefore f⁻¹(y)=−√y, with y≥0.
Challenge. Find the inverse of f(x)=1/(x−2) restricted to x>2.
y=1/(x−2) ⇒ x=2+1/y; since y>0, f⁻¹(y)=2+1/y for y>0.
Multiple choice. For f(x)=x² on x≥0, the range (and inverse domain) is which?
Correct answer: A. Squares are non-negative on x≥0, so the range of f becomes the domain of its inverse.
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