Maths Extension 1 Year 11 - Module 1 - Lesson 13
Graphical Relationships: Further Cases
Name
Date
Class
1. Lesson focus
Graphical relationship y=√f(x). The square-root transformation requires f(x)≥0; its range is non-negative and its graph starts at zeros of f.
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. State the domain of y=√(4−x).
3. State the domain of y=√(x²−9).
3. Challenge
For f(x)=x−1, find and describe y=[f(x)]².
4. Multiple choice
For y=√f(x), which condition defines the real-valued domain?
A. f(x)>1
B. f(x)≤0
C. f(x)≠0
D. f(x)≥0
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. State the domain and range of y=√(x−1).
Require x−1≥0, so domain x≥1; square root gives range y≥0.
2. State the domain of y=√(4−x).
4−x≥0, so x≤4; range is y≥0.
3. State the domain of y=√(x²−9).
x²−9≥0 ⇒ (x−3)(x+3)≥0; domain x≤−3 or x≥3.
Challenge. For f(x)=x−1, find and describe y=[f(x)]².
y=(x−1)²: an upward parabola with vertex (1,0), domain all real x and range y≥0. Squaring f differs from taking √f(x): negative values of f are allowed before squaring.
Multiple choice. For y=√f(x), which condition defines the real-valued domain?
Correct answer: D. A real square root requires a non-negative radicand.
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