IB Maths AA SL · Unit 2: Functions
Exam-style IB Maths AA SL function fundamentals 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.
The question bank covers these function fundamentals question types (number of questions in brackets):
The function $f$ is defined by $f(x) = 2x - 4$. Find an algebraic expression for $f^{-1}(x)$.
1. Set $y = 2x - 4$.
2. Swap variables: $x = 2y - 4$.
3. Rearrange: $2y = x + 4$.
4. Divide by $2$: $y = \frac{x+4}{2}$.
5. State using formal notation: $\mathbf{f^{-1}(x) = 0.5x + 2}$.
Let $f(x) = \frac{1}{2}x + 2$ and $g(x) = x^2 - 4x + 3$. Find a simplified expression for $(f \circ g)(x)$ and hence determine its exact range.
1. Substitute: $(f \circ g)(x) = \frac{1}{2}(x^2 - 4x + 3) + 2 = 0.5x^2 - 2x + 3.5$.
2. Find vertex $x$: $x = \frac{-(-2)}{2(0.5)} = 2$.
3. Find vertex $y$: $0.5(4) - 2(2) + 3.5 = 1.5$.
4. Deduce range: parabola opens up (leading coeff positive), so vertex is a minimum. Range: $\mathbf{y \geq 1.5}$.
Let $f(x) = \frac{x+5}{x-1}$ for $x \neq 1$. Find $f^{-1}(x)$ and briefly explain the geometric significance.
1. Swap variables: $x = \frac{y+5}{y-1}$.
2. Multiply: $x(y - 1) = y + 5$.
3. Expand: $xy - x = y + 5$.
4. Group $y$: $xy - y = x + 5$.
5. Factorise: $y(x - 1) = x + 5$.
6. Isolate: $\mathbf{f^{-1}(x) = \frac{x+5}{x-1}}$. Since $f = f^{-1}$, the function is self-inverse — its graph is symmetric about $y = x$.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Consider the function \(f(x) = -x^2 + 4x\), restricted to the domain \(0 \le x \le 4\).
Write down the range of \(f\).
Using the graph, explain why the inverse function \(f^{-1}(x)\) does not exist for this domain.
Let \(f(x) = 3^x - 2\).
Sketch the graph of \(f(x)\) for \(-2 \le x \le 2\) on a set of axes. Clearly label the \(y\)-intercept and the horizontal asymptote.
Write down the domain and range of \(f^{-1}(x)\).
Let \(f(x) = \sqrt{x+4}\) and \(g(x) = x - 2\).
Find the domain and range of \(f(x)\).
Sketch the graphs of \(f(x)\) and \(g(x)\) on the same grid.
Use algebra to find the exact coordinates of the point(s) of intersection.
There are 41 exam-style function fundamentals questions in the AA SL question bank (Paper 1: 32 · Paper 2: 9), graded 6 easy, 19 medium, 6 hard, 5 very hard, 5 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Both. In the bank, Paper 1: 32 · Paper 2: 9. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Open the AA SL 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.