2.2 Functions, domain and range questions for IB Maths AI

30 questions on Functions, domain and range, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 2 Functions
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	14	3	17
Paper 2	10	3	13
Every paper	24	6	30

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00024Paper 1 · HL only · 6 marks

The function $g$ is defined by $g (x) = x^{2} - 6 x + 11$ , for $x \leq k$ .

(a)
Write down the largest value of $k$ for which $g$ has an inverse.
1 mark
(b)
Using this value of $k$ , find $g^{- 1} (x)$ , stating its domain.
4 marks
(c)
Find $g^{- 1} (18)$ .
1 mark

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Question 2 · AI-00302Paper 2 · HL only · 15 marks

Sofia drives a hybrid car on a 350 km journey from Madrid to Valencia. When she drives at a constant speed of $v$ km h⁻¹, the fuel consumption of the car, in litres per 100 km, is modelled by

c (v) = 0.0018 v^{2} - 0.24 v + 13, 40 \leq v \leq 130.

Fuel costs 1.92 euros (EUR) per litre. The cost, in EUR, of the fuel for the whole journey when the consumption is $x$ litres per 100 km is $K (x) = 6.72 x$ .

(a)
Find the fuel consumption when Sofia drives at 100 km h⁻¹.
1 mark
(b)
Show that $(K \circ c) (v) = 0.012096 v^{2} - 1.6128 v + 87.36$ .
2 marks
(c)
Explain why $K (x) = 6.72 x$ .
1 mark
(d)
Find $\frac{d}{d v} ((K \circ c) (v))$ .
2 marks
(e)
Find the speed at which Sofia should drive to minimize the cost of the fuel for the journey, and write down this minimum cost.
3 marks
(f)
Sofia has budgeted 50 EUR for fuel. Find the range of speeds at which she can drive so that the fuel cost does not exceed her budget.
3 marks
(g)
The function $c$ has an inverse when its domain is restricted to $\frac{200}{3} \leq v \leq 130$ . Find an expression for $c^{- 1} (x)$ for this restricted domain.
3 marks

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Question 3 · AI-00026Paper 1 · HL only · 7 marks

Consider the functions $f (x) = \sqrt{x - 2}$ , for $x \geq 2$ , and $g (x) = 10 - x^{2}$ , for $x \in R$ .

(a)
Find $(g \circ f) (6)$ .
2 marks
(b)
Show that $(f \circ g) (x) = \sqrt{8 - x^{2}}$ .
1 mark
(c)
Find the largest possible domain of $f \circ g$ .
2 marks
(d)
Find the range of $f \circ g$ .
2 marks

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Question 4 · AI-00306Paper 2 · HL only · 17 marks

An agronomist in Kenya studies the effect of fertilizer on the yield of maize. When $x$ kg of fertilizer is applied per hectare, the yield, in tonnes per hectare, is modelled by

Y (x) = 6.2 + 0.05 x - 0.0001 x^{2}, 0 \leq x \leq 250.

The farmer sells the maize for 210 US dollars (USD) per tonne, so the revenue per hectare from a yield of $y$ tonnes per hectare is $R (y) = 210 y$ .

(a)
Find the yield when 100 kg of fertilizer is applied per hectare.
1 mark
(b)
Find an expression for $(R \circ Y) (x)$ .
2 marks
(c)
Fertilizer costs 1.60 USD per kg, and other costs are 400 USD per hectare. Show that the profit per hectare, in USD, is given by
$P (x) = - 0.021 x^{2} + 8.9 x + 902.$
2 marks
(d)
Find $P^{'} (x)$ .
2 marks
(e)
Find the amount of fertilizer per hectare that maximizes the profit, and find this maximum profit.
3 marks
(f)
Explain why $Y$ has an inverse function on the domain $0 \leq x \leq 250$ .
1 mark
(g)
Show that $Y^{- 1} (x) = 250 - 100 \sqrt{12.45 - x}$ .
4 marks
(h)
Find $Y^{- 1} (10)$ and interpret your answer in context.
2 marks

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Question 5 · AI-00466Paper 1 · 6 marks

During a cricket practice, Rohan hits a ball upwards. The height of the ball above the ground, in metres, $t$ seconds after it is hit is modelled by

h (t) = - 4.9 t^{2} + 14 t + 1.8.

(a)
Find $h (1)$ and interpret your answer in context.
2 marks
(b)
Find the time at which the ball hits the ground.
2 marks
(c)
Write down an appropriate domain for $h$ in this context.
2 marks

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Question 6 · AI-00629Paper 2 · 15 marks

A fountain in a park in Seoul shoots a jet of water from a nozzle. The height of the jet above the ground, in metres, is modelled by

h (x) = - 0.45 x^{2} + 2.7 x + 0.6,

where $x$ is the horizontal distance, in metres, from the nozzle. The jet lands on the ground.

(a)
Write down the height of the nozzle above the ground.
1 mark
(b)
Find the horizontal distance from the nozzle to the point where the jet lands.
2 marks
(c)
Hence write down an appropriate domain for $h$ .
2 marks
(d)
Find the range of $h$ for this domain.
3 marks
(e)
Find the set of values of $x$ for which the jet is at least $3$ m above the ground.
3 marks
(f)
Explain why $h$ does not have an inverse function on the domain in part (c).
2 marks
(g)
The engineer restricts the domain to $3 \leq x \leq 6.21$ so that $h$ has an inverse. Find $h^{- 1} (2)$ for this restricted domain.
2 marks

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