IB Maths AI HL AI 2.1 Functions and graphs questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Question 1

Paper 1Easy5 marks

A positioning device converts a control setting xx to an output f(x)f(x) Figure 1 shows its calibration curve. The rule is

f(x)=(x−3)2−2f(x)=(x-3)^2-2

on the branch shown. Both xx and f(x)f(x) are measured in device units.

01234567-202468101214x (device units)f(x) (device units)
(a)

Use Figure 1 to write down the range of ff

[1]
(b)

Find f−1(x)f^{-1}(x)

[1]
(c)

State the domain of f−1f^{-1}

[1]
(d)

Complete each statement, including the appropriate domain for xx

(i)

(f∘f−1)(x)=…(f\circ f^{-1})(x)=\dots

[1]
(ii)

(f−1∘f)(x)=…(f^{-1}\circ f)(x)=\dots

[1]

Question 2

Paper 1Easy5 marks

A local market sells honey in jars. Let qq be the quantity, in thousands of jars per week, and let pp be the price, in dollars per jar. Figure 1 shows the demand and supply models for this market.

0102030405060700102030405060708090q (thousands of jars per week)p (dollars per jar)Demand: pD = 78 - 1.2qSupply: pS = 12 + 0.8q
(a)

Write down the quantity at which the demand curve intersects the quantity axis.

[1]
(b)

Using graphing technology, find the market equilibrium. Give the quantity and price, with their units.

[3]
(c)

Write down the price at which the supply curve intersects the price axis.

[1]

Question 3

Paper 1Medium6 marks

During an 8-hour monitoring period, the water depths in two tanks are modelled by

f(t)=−0.25(t−4)2+10,g(t)=t+2,f(t)=-0.25(t-4)^2+10,\qquad g(t)=t+2,

where f(t)f(t) is the depth in Tank A, g(t)g(t) is the depth in Tank B, and tt is the time in hours since monitoring began. Depths are in centimetres, and 0≤t≤80\leq t\leq 8

(a)

Construct a function D(t)D(t) such that D(t)>0D(t)>0 means Tank A is deeper than Tank B. Give D(t)D(t) in expanded form.

[2]
(b)

Use a GDC to graph y=D(t)y=D(t) then sketch the graph for 0≤t≤80\leq t\leq 8 in your answer booklet. Label both axes with units and indicate the yy-intercept, turning point, tt-intercept and value at t=8t=8

[3]
(c)

Use your sketch to state which tank has the greater depth at t=7t=7

[1]

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