IB Maths AI HL · Unit 2: Functions
IB Maths AI HL Composite, Inverse and Transformations Questions
Exam-style IB Maths AI HL composite, inverse and transformations questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 17 questions
- Paper 1: 17
- 5 easy
- 5 medium
- 5 hard
- 2 starter
- 3 worked examples
Practise Composite, Inverse and Transformations questions →
AI HL formula booklet
What's examined in AI HL composite, inverse and transformations
The question bank covers these composite, inverse and transformations question types (number of questions in brackets):
- Inverse Functions (9)
- Function Transformations (6)
- Composite Functions (2)
Composite, Inverse and Transformations worked examples
Worked example 1: Evaluating Composite Functions · easy
Let $f(x) = 2x - 5$ and $g(x) = x^2 + 1$. Calculate the exact value of $(g \circ f)(3)$.
1. Identify the correct order of operations: $(g \circ f)(3)$ is equivalent to $g(f(3))$.
2. Evaluate the inner function first by substituting $x = 3$ into $f(x)$: $f(3) = 2(3) - 5$.
3. Calculate the result for $f(3)$: $6 - 5 = 1$.
4. Substitute this result into the outer function $g(x)$: $g(1) = (1)^2 + 1$.
5. Calculate the final value to get $2$.
Examiner tip: A very common mistake is confusing $(g \circ f)(x)$ with the product $g(x) \times f(x)$. Always work from the inside bracket outwards.
Worked example 2: Finding the Inverse of a Rational Function · medium
Let $h(x) = \frac{3x - 5}{2x + 1}$ for $x \neq -0.5$. Find an algebraic expression for the inverse function $h^{-1}(x)$.
1. Set $y = h(x)$ and swap the variables $x$ and $y$ to begin finding the inverse: $x = \frac{3y - 5}{2y + 1}$.
2. Multiply both sides by the denominator to clear the fraction: $x(2y + 1) = 3y - 5$.
3. Expand the brackets: $2xy + x = 3y - 5$.
4. Rearrange to group all terms containing $y$ on one side of the equation: $2xy - 3y = -x - 5$.
5. Factorise out the $y$: $y(2x - 3) = -x - 5$.
6. Divide to isolate $y$, giving the final inverse function: $h^{-1}(x) = \frac{-x - 5}{2x - 3}$.
Examiner tip: When finding the inverse of a rational function, collecting all the $y$ terms on one side so you can factorise $y$ out is the crucial algebraic step that students frequently miss.
Worked example 3: Constructing Transformed Functions · hard
The graph of $y = \ln(x)$ undergoes a sequence of three geometric transformations: it is translated horizontally by $3$ units to the right, stretched vertically by a scale factor of $2$, and finally reflected in the $x$-axis to form the new function $g(x)$. Find the expression for $g(x)$ and state its exact domain.
1. Apply the horizontal translation ($3$ units right) by replacing $x$ with $(x - 3)$: $y_1 = \ln(x - 3)$.
2. Apply the vertical stretch (scale factor $2$) by multiplying the entire function by $2$: $y_2 = 2\ln(x - 3)$.
3. Apply the reflection in the $x$-axis by multiplying the entire function by $-1$: $g(x) = -2\ln(x - 3)$.
4. Determine the new domain by setting the argument of the logarithm to be strictly positive: $x - 3 > 0$.
5. The final expression is $g(x) = -2\ln(x - 3)$ and the domain is $x > 3$.
Examiner tip: Order matters when applying transformations. Always build your transformed function step-by-step in the exact sequence given in the prompt to avoid sign and scaling errors.
FAQ
How many IB Maths AI HL composite, inverse and transformations questions are there?
There are 17 exam-style composite, inverse and transformations questions in the AI HL question bank (Paper 1: 17), graded 5 easy, 5 medium, 5 hard, 2 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is composite, inverse and transformations on Paper 1 or Paper 2?
In the question bank these questions are set as Paper 1 questions.
Where can I get the mark schemes?
Open the AI HL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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