IB Maths AI HL 2 Functions Questions

Practise IB Maths AI HL functions through shared-core and HL graphs, regression, transformations, equations and technology-supported models in context.

Exam points

Use function notation, domains, ranges, composition and inverse relationships to represent inputs, outputs and model mappings.
Construct and transform graphs, then interpret intercepts, extrema, symmetry, asymptotes, intersections and other defining features.
Select, fit and use function models to calculate predictions, thresholds, optimisation quantities and parameter values within a valid domain.
Interpret model parameters such as roots, vertices, amplitudes, periods, asymptotes, phase shifts, carrying capacity and half-life.
Use technology, logarithmic linearisation or regression to solve and compare models, interpreting gradients, residuals and uncertainty.
Evaluate assumptions, domain choices, extrapolation risk and contextual reasonableness using data, graph behaviour and units.

Covered syllabus points

2.1 Functions - SL content 2.2 Functions - AHL content

[Maximum number: 3]

The Voronoi diagram below shows four supermarkets represented by points with coordinates A(0,0), B(6,0), C(0,6) and D(2,2). The vertices X, Y, Z are also shown. All distances are measured in kilometres.

The equation of (XY) is y=2-x and the equation of (YZ) is y=0.5x+3.5.
Find the coordinates of X.

identifying the correct equations to use:

y=2-x \text { and } y=2 x-7

evidence of solving their correct equations or finding points of intersection graphically

(3,-1)

Note: Accept an answer expressed as " x=3, y=-1 ".

[Maximum number: 7]

Walkea Tree is a furniture store. They offer a 30 % discount on all chairs priced over 250 Canadian dollars (CAD).
A different furniture store, LuxeCraft, offer customers a reduction of 150 CAD on all chairs priced over 250 CAD .

(a)

Given that x is the original price of a chair and x>250,

[ 3 ]

(i)

write down a function f(x) that models the final price of a chair at Walkea Tree, after the discount is applied.

[ 1 ]

f(x)=0.7 x OR $f(x)=x-0.3 x \quad$ A1

(ii)

write down a function g(x) that models the final price of a chair at LuxeCraft, after the reduction is applied.

A third store, Opulent Furnishings, decides to offer both a 30 % discount and a reduction of 150 CAD on all their chairs priced over 250 CAD .

[ 2 ]

$g(x)=x-150 \quad$ A1

(b)

Describe the meaning of $(f \circ g)(x)$ , in context.

[ 2 ]

attempt to discuss the order in which the reduction/discount are applied
$(f \circ g)(x)$ describes when the 150 CAD reduction is applied before the 30 % discount.

(c)

State, with justification, whether $(f \circ g)(x)$ or $(g \circ f)(x)$ is better for the customer when they buy a chair with an original price of 450 CAD .

[ 2 ]

EITHER
evidence of solving $(f \circ g)(x)$ and $(g \circ f)(x)$ for a price at 450 CAD

OR
$(f \circ g)(x)=0.7 x-105$ and $(g \circ f)(x)=0.7 x-150$ seen

OR
stating that it is better for 30 % to be deducted from a higher price

OR
the numbers 165, 210 are seen

THEN
$(g \circ f)(x)$ is better for the customer

Note: Do not award R0A1.

[Maximum number: 5]

The function f is defined by $f(x)=4 \ln (2 x-3)$ , where $x>\frac{3}{2}$ .
The graph of y=f(x) is obtained from the graph of $y=\ln x$ by a sequence of three transformations.

(a)

Describe fully these transformations, including the order in which they occur.

[ 3 ]

vertical stretch scale factor 4

Note: The vertical stretch can occur in any order.

EITHER
translation $\binom{3}{0}$ followed by horizontal stretch scale factor $\frac{1}{2}$

OR
horizontal stretch scale factor $\frac{1}{2}$ followed by translation $\binom{1.5}{0}$

Note: Award A1A0 in either approach, for correct transformations, but incorrect order.

(b)

The graph of y=g(x) is shown.
The graph passes through the points (-2,-2),(-1,0),(0,2),(1,3),(2,5) and (3,6).

Figure for Question (b) — IB Maths AI HL

Find $(f \circ g)(2)$ .

[ 2 ]

$\quad g(2)=5$ $f(5)=7.78 \quad(=7.78364 \ldots, 4 \ln 7)$

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (i)

Question (ii)

Question (b)

Question (c)

Question 3

Question (a)

Question (b)