Maths Extension 1 Year 11 - Module 1 - Lesson 11
Graphical Relationships: $y = 1/f(x)$
Name
Date
Class
1. Lesson focus
Graphical relationship y=1/f(x). Reciprocation swaps zeros of f into vertical asymptotes; non-zero horizontal limits transform by reciprocation.
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−1)/(x+2), simplify 1/f(x) and state its exclusions.
3. For f(x)=x²−4, state where y=1/f(x) is positive and its vertical asymptotes.
3. Challenge
For f(x)=3+1/x, find 1/f(x) and its domain.
4. Multiple choice
A zero of f(x) usually becomes what on y=1/f(x)?
A. an x-intercept
B. a vertical asymptote
C. a removable point always
D. a horizontal asymptote
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. For f(x)=x−2, state the asymptotes of y=1/f(x).
Vertical asymptote x=2 (zero of f); horizontal asymptote y=0.
2. For f(x)=(x−1)/(x+2), simplify 1/f(x) and state its exclusions.
1/f=(x+2)/(x−1); x=−2 is excluded because f=0, and x=1 is excluded because f is undefined.
3. For f(x)=x²−4, state where y=1/f(x) is positive and its vertical asymptotes.
It is positive when x²−4>0, i.e. x<−2 or x>2; vertical asymptotes x=−2,2.
Challenge. For f(x)=3+1/x, find 1/f(x) and its domain.
1/f=x/(3x+1); exclude x=0 (f undefined) and x=−1/3 (f=0).
Multiple choice. A zero of f(x) usually becomes what on y=1/f(x)?
Correct answer: B. As f approaches zero, its reciprocal becomes unbounded unless a cancellation changes the local behavior.
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