Maths Extension 1 Year 11 - Module 1 - Lesson 14
Module Synthesis & Connections
Name
Date
Class
1. Lesson focus
Functions synthesis and connections. Combine inverse, composition and graph transformations while checking domains at every step.
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. Let f(x)=x² with x≥0 and g(x)=x−1. Find (f∘g)(x) and its domain.
3. For h(x)=|x−2|, solve h(x)=3.
3. Challenge
If f is one-to-one, explain why f⁻¹(f(4))=4 and evaluate it.
4. Multiple choice
For f(x)=2x+1, what is f⁻¹(x)?
A. (x−1)/2
B. (x+1)/2
C. 2x−1
D. 1−2x
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Let f(x)=2x+1. Find f⁻¹(x).
y=2x+1 ⇒ x=(y−1)/2, so f⁻¹(x)=(x−1)/2.
2. Let f(x)=x² with x≥0 and g(x)=x−1. Find (f∘g)(x) and its domain.
(f∘g)(x)=(x−1)²; require g(x)≥0 for f's restricted domain, so x≥1.
3. For h(x)=|x−2|, solve h(x)=3.
x−2=±3, so x=−1 or x=5.
Challenge. If f is one-to-one, explain why f⁻¹(f(4))=4 and evaluate it.
By the inverse identity f⁻¹(f(x))=x on f's domain; at x=4 the value is 4.
Multiple choice. For f(x)=2x+1, what is f⁻¹(x)?
Correct answer: A. Swap y=2x+1 and solve: x=(y−1)/2.
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