IB Maths AI SL · Unit 2: Functions
Exam-style IB Maths AI SL applications of functions questions with worked solutions. Start with the 5 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 applications of functions question types (number of questions in brackets):
A function is defined as $f(x) = 3x^2 - 5x + 2$. Calculate the exact value of $f(-4)$.
1. Substitute $x = -4$: $f(-4) = 3(-4)^2 - 5(-4) + 2$.
2. Evaluate the squared term: $(-4)^2 = 16$.
3. Combine: $3(16) + 20 + 2 = 48 + 20 + 2$.
4. State: $\mathbf{f(-4) = 70}$.
Let $g(x) = \frac{12}{x} + x$ for $x > 0$. Using your GDC, find the exact value(s) of $x$ for which $g(x) = 7$.
1. Set up: $\frac{12}{x} + x = 7$.
2. Graph $Y_1 = \frac{12}{x} + x$ and $Y_2 = 7$.
3. Find intersections with the GDC intersection tool.
4. Extract: $x = 3$ and $x = 4$.
5. State: $\mathbf{x = 3 \text{ and } x = 4}$.
Consider $f(x) = 2^x$ and $g(x) = 10 - 2x$. Find the coordinates of the point of intersection to 3 s.f.
1. Recognise $f(x) = g(x)$: $2^x = 10 - 2x$.
2. Enter $Y_1 = 2^x$ and $Y_2 = 10 - 2x$ on the GDC.
3. Use the intersection tool.
4. Read: $x = 2.0588\ldots$, $y = 5.8823\ldots$
5. Round to 3 s.f.: $\mathbf{(2.06,\ 5.88)}$.
A taxi company models the cost $C$ (in euros) as $C(d) = 3.50d + 4.00$, where $d$ is distance in km. Calculate the exact distance for a total cost of $\text{€}32.00$.
1. Set: $3.50d + 4.00 = 32.00$.
2. Subtract $4.00$: $3.50d = 28.00$.
3. Divide by $3.50$: $d = \frac{28.00}{3.50}$.
4. Evaluate: $d = 8$.
5. State: passenger travels exactly $\mathbf{8\text{ km}}$.
Daily cost to produce $x$ water bottles is $C(x) = 450 + 3.5x$; daily revenue is $R(x) = 12.5x$. Find the minimum number of bottles that must be sold to make a strictly positive profit.
1. Profit: $P(x) = R(x) - C(x)$.
2. Substitute: $P(x) = 12.5x - (450 + 3.5x) = 9x - 450$.
3. Set up: $9x - 450 > 0$.
4. Solve: $9x > 450 \implies x > 50$.
5. State: since $x$ is a whole number strictly greater than 50, the minimum is $\mathbf{51}$ bottles.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
The value of a specific industrial machine, \(V\), in euros, can be modelled as a linear function of time, \(t\), in years since it was purchased. The model is given by \(V(t) = 12\,500 - 1100t\).
Write down the initial purchase price of the machine.
Calculate the value of the machine exactly 4 years after purchase.
Find the exact time, \(t\), when the machine loses all of its value (\(V=0\)).
Two taxi companies charge different rates based on the distance travelled, \(d\), in kilometres. Company A models its cost by \(C_A(d) = 2.50d + 4.00\). Company B models its cost by \(C_B(d) = 1.80d + 7.50\).
Sketch both cost models on the grid provided below.
Find the exact distance \(d\) for which both taxi companies charge the same amount.
A company manufactures ceramic mugs. The total cost, \(C(x)\), in euros to produce \(x\) mugs is given by \(C(x) = 150 + 4.50x\). The revenue, \(R(x)\), generated by selling \(x\) mugs is given by \(R(x) = 8.50x\).
Sketch both the Cost and Revenue functions on the grid provided below.
Formulate an expression for the profit, \(P(x)\), in terms of \(x\).
Find the exact minimum number of mugs the company must sell to make a strictly positive profit.
All 19 applications of functions questions with mark schemes →
There are 19 exam-style applications of functions questions in the AI SL question bank (Paper 1: 10 · Paper 2: 9), graded 4 easy, 5 medium, 4 hard, 5 very hard, 1 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Both. In the bank, Paper 1: 10 · Paper 2: 9. Practise with your GDC — AI papers expect calculator methods throughout.
Open the AI 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.