Properties of Functions - IB Math AA Question Bank

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Properties of Functions

Practice exam-style IB Math AA questions for Properties of Functions, aligned with the syllabus and grouped by topic.

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Question 1

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

The points $A(-2,5)$ and $B(4,-1)$ lie on a straight line. The line $L$ is perpendicular to $AB$ and passes through the midpoint of $AB$ .

Find the gradient of $AB$ .

[2]

Find the equation of $L$ , giving your answer in the form $y=mx+c$ .

[3]

State the coordinates of the $x$ -intercept of $L$ .

[1]

Question 2

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

Let $f(x)=\sqrt{5-2x}+1$ .

Find the largest possible domain of $f$ .

[2]

State the range of $f$ .

[1]

Solve $f(x)=4$ .

[2]

Question 3

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

Let $f(x)=2x-3$ and $g(x)=x^2+1$ , for $x\in\mathbb{R}$ .

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

[2]

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

[2]

State whether $f\circ g=g\circ f$ . Justify your answer.

[1]

Question 4

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

The total cost, $C$ dollars, of a guided walking tour for $n$ people is modelled by a linear function. For $8$ people the cost is $146, and for$ 15 $people the cost is$ 237.

Determine a model for $C$ in terms of $n$ .

[3]

Another company models its cost by $D=9.5n+105$ . Find the number of people for which the two companies have the same cost.

[2]

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Question 5

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The graphs of $y=x^2-4x+1$ and $y=-x+3$ intersect at the points $P$ and $Q$ .

Form an equation in $x$ whose solutions give the $x$ -coordinates of $P$ and $Q$ .

[1]

Solve this equation for $x$ .

[2]

Find the coordinates of $P$ and $Q$ .

[2]

Question 6

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=-(x+1)^2+4$ , for $-4\le x\le 2$ .

Find the $x$ -intercepts of the graph of $f$ .

[2]

State the vertex and the maximum value of $f$ .

[2]

Sketch the graph of $f$ , showing the endpoints of the domain.

[2]

Question 7

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Consider the equation

e^{2x}-5e^x+6=0

Use the substitution $u=e^x$ to write the equation as a quadratic equation in $u$ .

[2]

Hence solve the original equation for $x$ .

[2]

Question 8

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=(x-1)^2+3$ , for $x\ge 1$ .

State the range of $f$ .

[1]

Find $f^{-1}(x)$ , stating its domain.

[3]

Solve $f^{-1}(x)=5$ .

[1]

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Question 9

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let

f(x)=\sin x+a\sin 2x+b\cos x

where $a,b\in\mathbb{R}$ and $x\in\mathbb{R}$ . The function $f$ is odd, and $f\left(\dfrac{\pi}{6}\right)=1$ .

Determine the value of $b$ .

[2]

Find the value of $a$ .

[2]

State the value of $f\left(-\dfrac{\pi}{6}\right)$ .

[1]

Question 10

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=\sqrt{7-2x}+1

where $x$ is in the largest possible real domain.

Write down the domain and range of $f$ .

[2]

Find an expression for $f^{-1}(x)$ , stating its domain.

[3]

Question 11

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A machine produces a component whose mass after $t$ minutes is modelled by

m(t)=1.5t+20,\quad t\geq 0

The energy, $E$ joules, required to heat a component of mass $m$ grams is modelled by

E(m)=0.04m^2+12

Find an expression for $(E\circ m)(t)$ .

[2]

Calculate the energy required when $t=10$ .

[1]

Find the time at which the energy required is $80$ joules.

[2]

Question 12

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=(x-2)^2-5,\quad x\geq 2

Write down the range of $f$ .

[1]

Find an expression for $f^{-1}(x)$ , stating its domain.

[3]

Solve $f^{-1}(x)=6.5$ .

[1]

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Question 13

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined for $x\in\mathbb{R}$ by

f(x)=a\sin x+b\cos x+c

It is known that $f$ is an even function, $f(0)=5$ and $f(\pi)=1$ .

Determine the values of $a$ , $b$ and $c$ .

[4]

Write down the range of $f$ .

[1]

Question 14

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=x^2-6x+5,\quad x\leq 3

Explain why $f$ has an inverse function.

[1]

Find $f^{-1}(x)$ , stating its domain.

[3]

Solve $f^{-1}(x)=-2$ .

[1]

Question 15

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=x^5+ax^3+bx^2+cx$ , where $a,b,c\in\mathbb{R}$ . The function $f$ is odd. Also, $f(1)=0$ and $f(2)=18$ .

Determine the value of $b$ .

[2]

Find the values of $a$ and $c$ .

[3]

Question 16

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=mx+c$ , where $m,c\in\mathbb{R}$ . The function $f$ is self-inverse, $f$ is not the identity function, and $f(3)=1$ .

Show that $m=-1$ .

[2]

Find $c$ and hence write down $f(x)$ .

[2]

Find the fixed point of $f$ .

[1]

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Question 17

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $f$ is defined by

f(x)=\frac{2x+3}{x-2},\quad x\ne 2

Find $f^{-1}(x)$ , stating its domain.

[3]

Hence state whether $f$ is self-inverse.

[1]

Find the fixed points of $f$ .

[2]

Question 18

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=x^2-4x+1$ , for $x\le 2$ .

Write $f(x)$ in the form $(x-h)^2+k$ and state the range of $f$ .

[2]

Find $f^{-1}(x)$ , stating its domain.

[3]

Find $f^{-1}(6)$ .

[1]

Question 19

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=x^3-4x+1,\quad -3\leq x\leq 3

Using your GDC, find the zeros of $f$ .

[3]

Find the coordinates of the local maximum and the local minimum of the graph of $y=f(x)$ .

[3]

Question 20

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Two quantities, $A$ and $B$ , are modelled by

A(t)=8e^{-0.18t}+2,\quad B(t)=0.5t+3

where $t$ is measured in hours and $0\leq t\leq 20$ .

Find $A(0)-B(0)$ .

[1]

Using your GDC, find the value of $t$ for which $A(t)=B(t)$ . Give your answer correct to 2 decimal places.

[3]

State the time interval during which $A(t)>B(t)$ . Give the endpoint correct to 2 decimal places.

[1]

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Question 21

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=\frac{2x-5}{x-2}

Write down the domain and range of $f$ .

[2]

Verify algebraically that $f$ is self-inverse.

[3]

Question 22

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The function $h$ is defined for $x\in\mathbb{R}$ by

h(x)=a\sin(2x)+b\cos x+cx

It is known that $h$ is an odd function, $h\left(\frac{\pi}{2}\right)=1$ and $h\left(\frac{\pi}{6}\right)=5$ .

Find the value of $b$ .

[1]

Find the value of $c$ .

[2]

Find the value of $a$ .

[2]

Write down $h\left(-\frac{\pi}{6}\right)$ .

[1]

Question 23

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A linear function $f$ is defined by $f(x)=ax+b$ , where $a,b\in\mathbb{R}$ . It is known that $f(2)=7$ and $f(6)=-1$ .

Determine the function $f$ .

I.

Find the values of $a$ and $b$ .

[3]

II.

Write down $f(x)$ .

[1]

The inverse function is denoted by $f^{-1}$ .

I.

Find $f^{-1}(x)$ .

[2]

II.

Solve $f(x)=f^{-1}(x)$ .

[2]

Let $P$ be the point where the graphs of $y=f(x)$ and $y=f^{-1}(x)$ intersect. Determine the coordinates of $P$ and justify why $P$ lies on the line $y=x$ .

[2]

Question 24

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=3-\sqrt{2x+4}$ , for $x\geq -2$ .

Consider the domain and range of $f$ .

I.

Write down the domain of $f$ .

[1]

II.

Write down the range of $f$ .

[1]

Find the inverse function.

I.

Find an expression for $f^{-1}(x)$ .

[3]

II.

State the domain and range of $f^{-1}$ .

[2]

Solve equations involving $f$ .

I.

Solve $f(x)=1$ .

[2]

II.

Solve $f(x)=x$ .

[2]

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Question 25

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The function $f$ is defined by
$f(x)=x+2$ for $-3\leq x\leq 1$ , and $f(x)=2x+1$ for $1<x\leq 4$ .

Consider the values and range of $f$ .

AI.

Write down $f(-3)$ .

[1]

AII.

Write down $f(4)$ .

[1]

AIII.

State the range of $f$ .

[2]

The function $f$ is one-to-one.

BI.

Find $f^{-1}(x)$ as a piecewise function.

[3]

BII.

State the domain of $f^{-1}$ .

[1]

Use your expression for $f^{-1}$ to solve $f^{-1}(x)=2$ .

[2]

Question 26

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=x-1$ , where $x\in\mathbb{R}$ , and let $g(x)=\sqrt{x+3}$ , where $x\geq -3$ .

Find the two composite functions and their domains.

I.

Find $(f\circ g)(x)$ and state its domain.

[2]

II.

Find $(g\circ f)(x)$ and state its domain.

[2]

Solve $(f\circ g)(x)=(g\circ f)(x)$ .

[3]

Consider the inverse of $f$ .

CI.

Find $f^{-1}(x)$ .

[1]

CII.

State whether $f\circ g$ and $g\circ f$ are the same function. Justify your answer.

[1]

Question 27

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A straight cable is to be installed above a curved roof. The cable passes through the points $A(0,18)$ and $B(60,42)$ , where coordinates are measured in metres. The height of the roof above the same horizontal axis is modelled by

r(x)=10+8\sin(0.06x),\quad 0\le x\le 60

The support strut is a straight line that passes through the point $C(45,62)$ and is perpendicular to the cable.

I.

Find the gradient of the cable.

[1]

II.

Find an equation for the height, $L(x)$ , of the cable.

[2]

A support strut is perpendicular to the cable and passes through the point $C(45,62)$ .

I.

Find an equation of the support strut.

[2]

II.

Find the coordinates of the point where the support strut meets the cable.

[2]

I.

Write down a function $d(x)$ for the vertical distance from the roof to the cable.

[2]

II.

Using your GDC, find the minimum vertical distance between the roof and the cable.

[2]

Question 28

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=4-\sqrt{18-3x}

where $x$ is in the largest possible real domain.

I.

Write down the domain of $f$ .

[1]

II.

Find the range of $f$ .

[2]

I.

Find an expression for $f^{-1}(x)$ , stating its domain.

[3]

II.

Find $f^{-1}(1)$ .

[1]

Find the fixed point of $f$ , giving your answer to three significant figures.

[3]

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Question 29

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A heating process is modelled by the function

T(t)=18+12(1-e^{-0.25t}),\quad 0\le t\le 12

where $T$ is the temperature in degrees Celsius after $t$ minutes. The quality score of the product at temperature $T$ is modelled by

Q(T)=100-0.5(T-28)^2

I.

Find an expression for $(Q\circ T)(t)$ .

[2]

II.

Calculate the quality score after $6$ minutes.

[2]

I.

Find the time at which the quality score is a maximum.

[2]

II.

Find the maximum quality score.

[2]

Find the interval of time for which the quality score is at least $98$ .

[2]

Question 30

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined for $x\in\mathbb{R}$ by

f(x)=x+\sqrt{x^2+9}

The function is one-to-one.

Write down the range of $f$ .

[1]

Find $f^{-1}(x)$ , stating its domain.

[3]

Hence solve $f(x)=10$ .

[1]

Question 31

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

For a real constant $a$ , the function $f$ is defined by

f(x)=\frac{ax+6}{3x+2},\quad x\neq -\frac{2}{3}

Determine the value of $a$ for which $f$ is self-inverse.

[4]

For this value of $a$ , state the domain and range of $f$ .

[2]

Question 32

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

For $a\in\mathbb{R}$ , define the function $f_a:\mathbb{R}\to\mathbb{R}$ by $f_a(x)=a-x$ . Also define $f_0(x)=-x$ and, for $n\in\mathbb{Z}^+$ , let $T_a^n$ denote applying $T_a$ repeatedly $n$ times, where $T_a=f_a\circ f_0$ .

I.

Show that $f_a$ is self-inverse.

[2]

II.

Find the fixed point of $f_a$ .

[2]

I.

Find $T_a(x)$ and hence write down $T_a^n(x)$ .

[3]

II.

Hence solve $T_a^n(x)=f_a(x)$ , giving your answer in terms of $a$ and $n$ .

[3]

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Question 33

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

A function $f$ is defined on a domain symmetric about $0$ . Define

E(x)=\frac{f(x)+f(-x)}{2},\qquad O(x)=\frac{f(x)-f(-x)}{2}

The functions $E$ and $O$ are called the even and odd parts of $f$ .

I.

Show that $E$ is an even function.

[2]

II.

Show that $O$ is an odd function.

[2]

For $f(x)=e^x$ , find $E(x)$ and $O(x)$ .

[2]

I.

Solve $O(x)=3$ exactly.

[2]

II.

Explain why $E$ has no inverse function on $\mathbb{R}$ , but has an inverse function if its domain is restricted to $x\ge 0$ .

[2]

Question 34

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

Two linear functions are defined by $f(x)=mx+c$ and $g(x)=px+q$ , where $m,p,c,q\in\mathbb{R}$ . This question investigates when the order of two linear calibrations matters.

I.

Find $(f\circ g)(x)$ and $(g\circ f)(x)$ .

[2]

II.

Show that $f\circ g=g\circ f$ if and only if $c(p-1)=q(m-1)$ .

[2]

Now let $f(x)=2x+c$ and $g(x)=px+3$ , where $p\ne 1$ . If $f\circ g=g\circ f$ , express $c$ in terms of $p$ .

[2]

I.

Suppose also that $g$ is self-inverse. Find $p$ and $c$ .

[3]

II.

Find the common fixed point of $f$ and $g$ for these values.

[1]

Question 35

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

Let $a>0$ . Define

f(x)=\sqrt{x+a},\qquad x\ge -a

and

g(x)=x^2-a,\qquad x\in\mathbb{R}

The two functions appear to undo each other, but only in one order without further restriction.

I.

State the domain and range of $f$ .

[2]

II.

Show that $(g\circ f)(x)=x$ for all $x$ in the domain of $f$ .

[2]

Find $(f\circ g)(x)$ and state its largest possible domain.

[2]

I.

Restrict the domain of $g$ so that $g^{-1}=f$ .

[2]

II.

Explain why $(f\circ g)(x)=x$ only on this restricted domain.

[2]

Question 36

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

Let $a>0$ and define

f(x)=\sqrt{a^2-x^2}

The largest possible real domain is used unless otherwise stated.

I.

State the domain and range of $f$ .

[2]

II.

Show that $f$ is even, and explain why it has no inverse function on its largest domain.

[2]

Restrict the domain to $0\le x\le a$ . Find $f^{-1}(x)$ , stating its domain.

[3]

On the restricted domain $0\le x\le a$ , find the intersection of $y=f(x)$ and $y=\frac{x}{2}$ .

[2]

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Question 37

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

A function on a finite set can be represented by a mapping table. Let $S=\{1,2,3,4\}$ . Functions $f:S\to S$ and $g:S\to S$ are given by the table.

x	f(x)	g(x)
1	3	2
2	4	1
3	1	4
4	2	3

I.

Verify that $f$ is self-inverse.

[2]

II.

State whether $g$ is self-inverse.

[1]

Let $h=f\circ g$ . Copy and complete the mapping table for $h$ .

[3]

I.

Determine whether $h$ is self-inverse.

[2]

II.

Find the fixed points of $h$ .

[1]

Question 38

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Let $f(x)=x^2-2x-3$ and $g(x)=x+1$ , for $x\in\mathbb{R}$ . The function $d$ is defined by $d(x)=f(x)-g(x)$ .

Consider the difference function $d$ .

I.

Find $d(x)$ .

[1]

II.

Solve $d(x)=0$ .

[3]

Find the coordinates of the intersection points of the graphs of $y=f(x)$ and $y=g(x)$ .

[2]

Analyse the vertical separation of the two graphs.

I.

Write $d(x)$ in completed-square form and state its minimum value.

[2]

II.

Hence state the value of $x$ for which $g(x)-f(x)$ is greatest, and state this greatest value.

[2]

Question 39

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Consider the function $h$ defined by $h(x)=e^{2x}-6e^x+8$ , for $x\in\mathbb{R}$ .

Use the substitution $u=e^x$ .

I.

Using the substitution $u=e^x$ , write $h(x)=0$ as a quadratic equation in $u$ .

[1]

II.

Solve this quadratic equation for $u$ .

[2]

Hence solve $h(x)=0$ .

[2]

Solve the inequality $h(x)\leq 0$ .

[3]

The equation $h(x)=0$ can be interpreted as the intersection of the graphs $y=e^x$ and $y=6-8e^{-x}$ . Find the coordinates of these intersection points.

[2]

Question 40

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $f(x)=ax^4+bx^3+cx^2+dx+e$ , where $a,b,c,d,e\in\mathbb{R}$ . The function $f$ is even. It is also known that $f(0)=3$ , $f(1)=0$ and $f(2)=15$ .

Part (a): Determine the constants in $f$ .

I.

Use the fact that $f$ is even to state the values of $b$ and $d$ .

[2]

II.

Find $a$ , $c$ and $e$ .

[4]

Part (b): Define the function by $g(x)=xf(x)$ .

I.

Show that $g$ is an odd function.

[2]

II.

Find all zeros of $g$ .

[2]

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Question 41

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For a real constant $a$ , the function $f$ is defined by $f(x)=(x-a)^2+1$ , for $x\geq a$ . It is known that $f(5)=10$ .

Determine the value of $a$ and the range of $f$ .

I.

Show that $a=2$ .

[3]

II.

State the range of $f$ .

[1]

Use $a=2$ for this part. If you did not obtain $a=2$ , use $f(x)=(x-2)^2+1$ , $x\geq 2$ .

I.

Find $f^{-1}(x)$ , stating its domain.

[3]

II.

Solve $f^{-1}(x)=x$ .

[3]

Question 42

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $h(x)=x^3-4x^2+x+6$ , for $x\in\mathbb{R}$ . Define
$E(x)=\dfrac{h(x)+h(-x)}{2}$ and $O(x)=\dfrac{h(x)-h(-x)}{2}$ .

Find $E(x)$ and $O(x)$ .

I.

Write down $h(-x)$ .

[1]

II.

Find $E(x)$ and $O(x)$ .

[3]

Consider the symmetry of $E$ and $O$ .

I.

Show that $E$ is even and $O$ is odd.

[2]

II.

Verify that $h(x)=E(x)+O(x)$ .

[1]

Solve $h(x)=h(-x)$ .

[2]

Find the zeros of $E$ and explain their symmetry.

[2]

Question 43

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $f(x)=\sqrt{x-1}$ , where $x\geq 1$ , and let $g(x)=3x-2$ , where $x\in\mathbb{R}$ . Define $h=f\circ g$ .

Find the function $h$ .

I.

Find an expression for $h(x)$ .

[1]

II.

State the domain and range of $h$ .

[2]

Find $h^{-1}(x)$ directly, stating its domain.

[3]

Use the inverse functions of $f$ and $g$ .

I.

Find $f^{-1}(x)$ and $g^{-1}(x)$ .

[2]

II.

Verify that $h^{-1}=g^{-1}\circ f^{-1}$ .

[1]

Hence evaluate $h^{-1}(3)$ and explain its meaning in terms of $h$ .

[1]

Question 44

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

The height, $y$ metres, of a small drone above level ground after $x$ seconds is modelled by

y=f(x)=x^3-6x^2+9x+2,\quad 0\le x\le 5

A warning signal is triggered whenever the drone is above $4$ metres.

I.

Find the coordinates of the local maximum point of the graph of $f$ .

[2]

II.

Find the coordinates of the local minimum point of the graph of $f$ .

[2]

Find the times at which the warning signal changes state.

[3]

I.

State the intervals of time during which the warning signal is on.

[2]

II.

Find the total area between the graph of $f$ and the line $y=4$ over the intervals when the warning signal is on.

[3]

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Question 45

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

Consider the function

f(x)=x^3-3x+2

The domain of $f$ is first taken to be $-1\le x\le 1$ .

I.

Find the range of $f$ on this domain.

[2]

II.

Explain why $f$ has an inverse function on this domain.

[1]

I.

State the domain and range of $f^{-1}$ .

[2]

II.

Using your GDC, find $f^{-1}(1.5)$ .

[2]

I.

Sketch the graphs of $y=f(x)$ and $y=f^{-1}(x)$ on the same axes.

[2]

II.

Find the point of intersection of the graphs of $y=f(x)$ and $y=f^{-1}(x)$ .

[3]

Question 46

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

Let

h(x)=\cos x+0.2x,\quad 0\le x\le 10

The graph of $y=h(x)$ is used to solve equations and inequalities.

I.

Using your GDC, find all zeros of $h$ .

[2]

II.

State the solution of $h(x)<0$ .

[2]

I.

Find the minimum value of $h(x)$ on $0\le x\le 10$ .

[2]

II.

Explain why there are no zeros of $h$ for $5<x\le 10$ .

[1]

Find the area enclosed by the graph of $h$ and the $x$ -axis.

[3]

Question 47

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The function $f$ is defined for $x\in\mathbb{R}$ by

f(x)=a\sin(2x)+b\cos x+c\cos(2x)+d

where $a,b,c,d\in\mathbb{R}$ . It is known that $f$ is even, $f(0)=5$ , $f(\pi)=1$ and $f\left(\frac{\pi}{3}\right)=1$ .

I.

Show that $a=0$ .

[2]

II.

Find the values of $b$ , $c$ and $d$ .

[3]

For the rest of the question, take $f(x)=2\cos x+2\cos(2x)+1$ .

I.

Find the range of $f$ .

[3]

II.

Find all solutions of $f(x)=1$ for $0\le x\le 2\pi$ .

[1]

The domain of $f$ is restricted to $0\le x\le \pi$ . Explain why this restricted function does not have an inverse function.

[3]

Question 48

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The function $f$ is defined by

f(x)=x^3-3x^2+2

First consider $f$ on the domain $x\in\mathbb{R}$ .

I.

Explain why $f$ does not have an inverse function on $\mathbb{R}$ .

[2]

II.

The domain is now restricted to $x\ge2$ . State the range of this restricted function.

[2]

Let $g$ be the inverse function of the restriction of $f$ to the domain $x\ge2$ .

I.

State the domain and range of $g$ .

[2]

II.

Using your GDC, find $g(10)$ .

[2]

Find the gradient of the graph of $y=g(x)$ at the point where $x=10$ .

[3]

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Question 49

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The function $f$ is defined by

f(x)=x^4-4x^2

I.

Show that $f$ is even.

[2]

II.

Explain why $f$ does not have an inverse function on $\mathbb{R}$ .

[2]

For parts (b) and (c), restrict the domain of $f$ to $x\ge\sqrt{2}$ .

I.

State the range of this restricted function.

[2]

II.

Find $f^{-1}(x)$ , stating its domain.

[3]

Find the exact value of $f^{-1}(12)$ and verify your answer by composition.

[3]

Question 50

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For real constants $p$ and $q$ , consider

f(x)=\frac{px+q}{x-p},\qquad x\ne p

You may assume $q\ne -p^2$ .

I.

Find $f^{-1}(x)$ .

[3]

II.

Explain the role of the condition $q\ne -p^2$ .

[2]

For $p=2$ and $q=5$ , state the equations of the asymptotes and find the intercepts with the axes.

[3]

I.

For $p=2$ and $q=5$ , find the fixed points of $f$ .

[2]

II.

Justify geometrically why the graph is symmetric in the line $y=x$ .

[2]

Question 51

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let $a>0$ . The function $f$ is defined by

f(x)=x^2-2ax,\qquad x\ge a

This is one branch of a parabola.

Object	Equation	Domain / condition
Branch of $f$	$y=x^2-2ax$	$a>0$ , $x\ge a$
Line	$y=x$	All real $x$

I.

State the vertex of the graph and the range of $f$ .

[3]

II.

Find $f^{-1}(x)$ , stating its domain.

[3]

Find, in terms of $a$ , the intersection point of the graph of $f$ and the line $y=x$ .

[2]

Given that the minimum value of $f$ is $-9$ , find the intersection point of the graph of $f$ and the line $y=x$ .

[2]

Question 52

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For a real parameter $k$ , define

h_k(x)=e^{-x}-x-k,\qquad x\ge 0

The equation $h_k(x)=0$ may be interpreted as the intersection of $y=e^{-x}$ and $y=x+k$ on the restricted domain $x\ge 0$ .

I.

Show that $h_k$ is strictly decreasing on $x\ge 0$ .

[2]

II.

Determine the values of $k$ for which $h_k(x)=0$ has a solution on $x\ge 0$ .

[3]

I.

Use your GDC to solve $e^{-x}=x+0.2$ for $x\ge 0$ .

[2]

II.

Find the value of $k$ for which the solution is $x=0.7$ .

[3]

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Question 53

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $a\ne 0$ , define

f_a(x)=\frac{a}{x},\qquad x\ne 0

This question investigates a simple family of self-inverse functions and their intersections with straight lines through the origin.

I.

Verify that $f_a$ is self-inverse.

[2]

II.

Determine whether $f_a$ is odd, even or neither.

[2]

Let $m\ne 0$ . Determine the condition on $a$ and $m$ for the graphs of $y=f_a(x)$ and $y=mx$ to have real intersections.

[3]

I.

For $a=12$ and $m=3$ , find the intersection points.

[2]

II.

State the fixed points of $f_{12}$ .

[1]

Question 54

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $a\in\mathbb{R}$ , define

f_a(x)=(x-a)^3+a,\qquad x\in\mathbb{R}

The graph is a cubic translated so that its centre of rotational symmetry is $(a,a)$ .

I.

Find $f_a^{-1}(x)$ .

[2]

II.

Explain why $f_a$ is one-to-one on $\mathbb{R}$ .

[1]

Find the fixed points of $f_a$ , giving your answers as coordinates.

[3]

I.

Determine the value of $a$ for which the three fixed points have $x$ -coordinates with product $6$ .

[2]

II.

For this value of $a$ , use your GDC to confirm the intersections of $y=f_a(x)$ and $y=x$ .

[2]

Question 55

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For $k\in\mathbb{R}$ , define $F_k(x)=\dfrac{kx+1}{x-k}$ , where $x\neq k$ .

Consider the case $k=2$ .

I.

State the domain of $F_2$ .

[1]

II.

Show that $2$ is not in the range of $F_2$ .

[1]

Use composition to examine $F_2$ .

I.

Show that $(F_2\circ F_2)(x)=x$ .

[3]

II.

Hence write down $F_2^{-1}(x)$ .

[1]

Find the fixed points of $F_2$ .

[3]

Show that $F_k$ is self-inverse for every real value of $k$ .

[3]

Question 56

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The function $f$ is defined by $f(x)=\ln\left(\dfrac{1+x}{1-x}\right)$ , for $-1<x<1$ .

Consider the symmetry of $f$ .

I.

Show that $f$ is an odd function.

[3]

II.

State the symmetry of the graph of $y=f(x)$ .

[1]

Find the inverse function.

I.

Find $f^{-1}(x)$ .

[4]

II.

State the domain and range of $f^{-1}$ .

[2]

Use the inverse function.

I.

Solve $f(x)=\ln 3$ .

[1]

II.

Show that $f^{-1}$ is an odd function.

[1]

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Question 57

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For a real parameter $a$ , define

f_a(x)=\frac{2x+a}{ax-2},\quad x\ne \frac{2}{a},\ a\ne0

I.

State the domain of $f_a$ .

[1]

II.

Verify algebraically that $f_a$ is self-inverse.

[3]

It is given that $f_a(1)=5$ .

I.

Using the given condition $f_a(1)=5$ , find the value of $a$ .

[2]

II.

Hence write down $f_a^{-1}(5)$ .

[1]

For $a=3$ , find the fixed points of $f_a$ and determine the set of values of $x$ for which $f_a(x)>x$ .

[5]

Question 58

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Consider the function

h(x)=2\cos x+\sin^2 x,\quad -\pi\le x\le \pi

I.

Show that $h$ is an even function.

[2]

II.

Explain why $h$ does not have an inverse function on $[-\pi,\pi]$ .

[1]

Let $g$ be the restriction of $h$ to the domain $0\le x\le\pi$ .

I.

Find the range of the restricted function.

[2]

II.

Find an expression for $h^{-1}(x)$ , stating its domain.

[3]

I.

Find $g^{-1}(0)$ .

[2]

II.

Using your GDC, find the intersections of the graphs of $y=h(x)$ and $y=g^{-1}(x)$ .

[2]

Question 59

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The function $f$ is defined for all real $x$ by

f(x)=x^3+4x

Let $g=f^{-1}$ .

I.

Show that $f$ is odd.

[2]

II.

Explain why $f$ has an inverse function.

[2]

I.

Using your GDC, find $g(20)$ .

[2]

II.

Hence find $g(-20)$ , giving a reason.

[2]

Solve $f(f(x))=10$ .

[4]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $a\in\mathbb{R}$ , define

f_a(x)=\cos x+a\cos 2x,\qquad x\in\mathbb{R}

The graph is periodic. This question investigates its symmetry and range by using $u=\cos x$ .

I.

Show that $f_a$ is even.

[1]

II.

Using $u=\cos x$ , express $f_a(x)$ as a quadratic in $u$ .

[2]

For $a=1$ , determine the range of $f_1$ .

[4]

I.

Determine the value of $a$ for which the quadratic in $u$ has its vertex at $u=-\frac{1}{2}$ .

[2]

II.

For this value of $a$ , solve $f_a(x)=0$ for $0\le x<2\pi$ .

[3]

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