Maths Extension 1 Year 11 - Module 1 - Lesson 5
Introduction to Inverse Functions
Name
Date
Class
1. Lesson focus
Inverse functions and composition checks. To find an inverse, write y=f(x), swap x and y, solve for y, and check both compositions on their domains.
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. Verify f(f⁻¹(x)) for f(x)=2x−5.
3. Find the inverse of g(x)=(x−1)/3.
3. Challenge
For h(x)=4−x, find h⁻¹ and state whether it equals h.
4. Multiple choice
If f is one-to-one and f(a)=b, then f⁻¹(b) equals what?
A. 1/a
B. b
C. f(b)
D. a
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Find the inverse of f(x)=2x−5.
y=2x−5 ⇒ x=(y+5)/2; hence f⁻¹(x)=(x+5)/2.
2. Verify f(f⁻¹(x)) for f(x)=2x−5.
2((x+5)/2)−5=x+5−5=x.
3. Find the inverse of g(x)=(x−1)/3.
y=(x−1)/3 ⇒ x=3y+1; hence g⁻¹(x)=3x+1.
Challenge. For h(x)=4−x, find h⁻¹ and state whether it equals h.
Swap and solve: y=4−x gives x=4−y, so h⁻¹(x)=4−x=h(x).
Multiple choice. If f is one-to-one and f(a)=b, then f⁻¹(b) equals what?
Correct answer: D. The inverse reverses each input-output pair: f(a)=b implies f⁻¹(b)=a.
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