IB myp 4-5 MATHEMATICS – Practice Questions- All Topics
Topic :Functions–Rational functions
Topic :Function- Weightage : 21 %
All Questions for Topic : Representation and shape of more complex functions,Transformation of quadratic functions,Rational functions,Graphing trigonometric functions,Linear programming, including inequalities,Networks-edges and arcs, nodes/ vertices, paths,Calculating network pathways,Weighted networks,Domain and range
Question : Function Composition and Inverses [12 marks]
Given the functions:
f(x) = (x – 2)/(3x – 11)
g(x) = x + 3
a Question a [3 marks] – Function Composition
Find f∘g(x) and simplify your answer.
Show Solution
Step 1: Substitute g(x) into f(x)
f∘g(x) = f(g(x)) = f(x + 3) = [(x + 3) – 2]/[3(x + 3) – 11]
Step 2: Simplify numerator and denominator
Numerator: (x + 3 – 2) = (x + 1)
Denominator: (3x + 9 – 11) = (3x – 2)
Final simplified form:
f∘g(x) = (x + 1)/(3x – 2) AG (Answer Given)
Marking Notes:
Award marks for substitution and each simplification step
b Question b [5 marks] – Inverse Function
Find the inverse function (f∘g)-1(x) and state its domain.
Show Solution
Step 1: Set y = f∘g(x)
y = (x + 1)/(3x – 2)
Step 2: Swap x and y and solve for y
x = (y + 1)/(3y – 2)
Step 3: Cross-multiply
x(3y – 2) = y + 1
3xy – 2x = y + 1
Step 4: Collect y terms
3xy – y = 2x + 1
y(3x – 1) = 2x + 1
Step 5: Solve for y
y = (2x + 1)/(3x – 1)
Final inverse function:
(f∘g)-1(x) = (2x + 1)/(3x – 1)
Domain restriction:
x ≠ 1/3 (since denominator cannot be zero)
Alternative Notation:
Domain can be written as ℝ\{1/3} or x ∈ ℝ, x ≠ 1/3
c Question c Alternative Method [4 marks]
Using an alternative approach, verify your answer for part (b) by finding h and k such that (f∘g)(x) can be written in the form (x – h)/k + c.
Show Alternative Solution
Step 1: Rewrite f∘g(x) in alternative form
(x + 1)/(3x – 2) = [1/3(3x – 2) + 5/3]/(3x – 2)
= 1/3 + (5/3)/(3x – 2)
Step 2: Identify transformations
This shows the function has:
- Vertical shift: c = 1/3
- Horizontal shift: h = 2/3
- Vertical stretch: k = 5/9
Step 3: Find inverse using transformations
The inverse would have:
x = 1/3 + (5/3)/(3y – 2)
Solving this leads to the same inverse function:
y = (2x + 1)/(3x – 1)
Verification:
Both methods yield the same inverse function, confirming correctness
Syllabus Reference
Unit 3: Function
- Function composition
- Inverse functions
- Rational functions
Unit 2: Algebra
- Algebraic manipulation
- Solving equations
Assessment Criteria: B (Investigating patterns) and C (Communicating mathematics)