Properties of Functions - IB Math AI Question Bank

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Properties of Functions

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

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Question 1

SL • Paper 1

Easy

Calculator Permitted

SL • Paper 1

Easy

Calculator Permitted

The real-valued function $f$ is defined by

f(x)=\sqrt{12-3x}+2

Calculate $f(-1)$ .

[1]

Determine the domain of $f$ .

[2]

State the range of $f$ .

[1]

Question 2

SL • Paper 1

Easy

Calculator Permitted

SL • Paper 1

Easy

Calculator Permitted

A temperature of $F$ degrees Fahrenheit is converted to $C$ degrees Celsius using

C(F)=\frac{5}{9}(F-32)

Find the Celsius temperature corresponding to $95^\circ\text{F}$ .

[1]

Find the Fahrenheit temperature corresponding to $18^\circ\text{C}$ .

[2]

Explain how the calculation in part (b) is related to the inverse of $C$ .

[1]

Question 3

HL • Paper 1

Easy

Calculator Permitted

HL • Paper 1

Easy

Calculator Permitted

A delivery route has length $m$ miles, where $0\leq m\leq 300$ . The function

f(m)=1.609m

converts the distance to kilometres. The delivery charge for a route of $k$ kilometres is modelled by

g(k)=0.12k+4

where $g(k)$ is measured in dollars.

Find an expression for $(g\circ f)(m)$ .

[2]

Calculate the delivery charge for a route of $85$ miles.

[1]

State the domain and range of $g\circ f$ in this context.

[2]

Question 4

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

The function $g$ is defined by

g(x)=x^3-4x^2-x+4,\qquad -2\leq x\leq 5

Determine the $x$ -intercepts of the graph of $g$ .

[2]

Find the coordinates of the local minimum point of $g$ .

[2]

State the maximum value of $g$ on the given domain.

[1]

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Question 5

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

For a particular product, the demand price $D(q)$ and supply price $S(q)$ , in dollars per item, are modelled by

D(q)=80e^{-0.08q}+10

and

S(q)=2.5q+8

where $q$ is the number of hundreds of items produced. Market equilibrium occurs when the demand price equals the supply price.

Find the point of intersection of the graphs of $D$ and $S$ .

[3]

Interpret the first coordinate of this point in context.

[1]

Question 6

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

The functions $f$ and $g$ are defined by

f(x)=\sqrt{x+2},\qquad g(x)=\frac{1}{x-1}

The difference function $d$ is defined by $d(x)=f(x)-g(x)$ .

Write down an expression for $d(x)$ .

[1]

Determine the domain of $d$ .

[2]

Calculate $d(2)$ .

[1]

Question 7

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

A continuous function $h$ has domain $-3\leq x\leq 5$ . Its graph has the following features:

endpoints $(-3,2)$ and $(5,-\frac{1}{2})$ ;
$x$ -intercepts $(-2,0)$ and $(2,0)$ ;
a local minimum at $(-2,0)$ ;
a local maximum at $(0,4)$ ;
a local minimum at $(4,-1)$ .

The graph has no other turning points or intercepts.

Sketch the graph of $h$ on the axes provided. Label all intercepts and turning points.

[4]

Hence, state the range of $h$ .

[2]

Question 8

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

The function $h$ is defined by

h(x)=\frac{2x+3}{x-4},\qquad x\neq 4

Find the coordinates of the $x$ -intercept and the $y$ -intercept of the graph of $h$ .

[2]

State the equations of the vertical and horizontal asymptotes of the graph of $h$ .

[2]

State the range of $h$ .

[1]

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Question 9

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The function $q$ is defined by

q(x)=(x-2)^2+5

with domain $x\geq 2$ .

Find $q^{-1}(x)$ and state its domain.

[3]

Calculate $q^{-1}(30)$ .

[1]

If the domain of $q$ were instead restricted to $x\leq2$ , state the resulting inverse function and calculate its value at $30$ .

[2]

Question 10

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A temperature sensor converts a temperature $T$ , in degrees Celsius, into a voltage $V(T)$ , in volts, using

V(T)=0.02T+1.5

where $-20\leq T\leq80$ . A digital converter then changes a voltage $v$ into a reading $R(v)$ using

R(v)=409.6v

Find an expression for the digital reading as a function of temperature.

[2]

A digital reading of $860$ is recorded. Determine the corresponding temperature.

[2]

Question 11

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

The height of a weather balloon above the ground, $h(t)$ metres, is modelled for $0\leq t\leq 14$ by

h(t)=-0.04t^3+0.48t^2+1.2t+2

where $t$ is the time in minutes after its release.

Determine when the balloon reaches its maximum height.

[2]

Find the maximum height of the balloon.

[1]

Calculate the height of the balloon at the end of the modelled interval.

[1]

State the range of $h$ over the modelled interval.

[2]

Question 12

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=\frac{3x-2}{x+4},\qquad x\neq -4

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

[3]

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

[1]

Verify that $(f^{-1}\circ f)(x)=x$ on the domain of $f$ .

[2]

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Question 13

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The functions $f$ and $g$ are defined by

f(x)=2x+1,\qquad x\in\mathbb{R}

and

g(x)=x^2,\qquad x\geq0

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

[2]

Solve $(f\circ g)(x)=(g\circ f)(x)$ , taking account of the domains of both composite functions.

[3]

Question 14

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=\ln(x-1)+3

with domain $x>1$ .

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

[3]

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

[2]

Verify that $(f\circ f^{-1})(x)=x$ .

[1]

Question 15

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The linear function $f$ is defined by

f(x)=ax+b

where $a\neq0$ . Its inverse is

f^{-1}(x)=0.25x-3

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

[3]

Verify both inverse composition identities for these functions.

[2]

Question 16

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The function $p$ is defined by

p(x)=3-(x+1)^2

with domain $x\leq-1$ .

Explain why the stated domain allows $p$ to have an inverse function.

[1]

Find $p^{-1}(x)$ and state its domain.

[3]

Hence, solve $p(x)=-13$ on the stated domain.

[2]

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Question 17

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The depth of water, $f(t)$ metres, in a drainage channel is modelled by

f(t)=\sqrt{36-4t}+1

where $t$ is the time in hours after a gate is opened and $0\leq t\leq9$ . A second channel has depth $g(t)=0.5t+2$ metres over the same interval.

I.

Calculate the initial depth and the depth after $9$ hours in the first channel.

[2]

II.

State the domain and range of $f$ in this context.

[2]

Determine when the two channels have the same depth, and state this common depth.

[2]

Determine the length of time for which the depth in the first channel is at least $4$ metres.

[2]

Question 18

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The concentration of a nutrient in a tank is modelled by

p(t)=8+\frac{12t}{t+3},\qquad 0\leq t\leq30

where $p(t)$ is measured in milligrams per litre and $t$ is measured in minutes.

I.

Calculate $p(0)$ and $p(30)$ .

[2]

II.

State the equation of the horizontal asymptote of the unrestricted graph of $p$ .

[2]

Determine the time at which the concentration first reaches $17$ milligrams per litre.

[2]

Explain why the model never predicts a concentration of $20$ milligrams per litre at a finite time.

[2]

Question 19

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The strength of a wireless signal at horizontal distance $d$ metres from a transmitter is modelled by

S(d)=\frac{100}{d^2+4},\qquad 0\leq d\leq20

where $S(d)$ is measured in arbitrary signal units.

I.

Find the maximum and minimum values of $S$ on the stated domain.

[2]

II.

Hence state the range of $S$ in this context.

[2]

Find $S^{-1}(2)$ and interpret your answer in context.

[2]

Explain why $S^{-1}$ exists when $S$ has the stated domain, but would not exist if the domain were $-20\leq d\leq20$ .

[2]

Question 20

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

A monitoring system records the time separation from an event at $t=4$ seconds using

x(t)=|t-4|

where $0\leq t\leq10$ . A detector converts this separation into a signal

$q(u)=3u+2$

I.

Find the signal function $S=q\circ x$ and state its range.

[2]

II.

Explain why $S$ has no inverse function on $0\leq t\leq10$ .

[2]

I.

Restrict the domain to $4\leq t\leq10$ . Find the inverse of the restricted signal function.

[2]

II.

A signal of $14$ is recorded during the restricted interval. Determine the time of the recording.

[2]

State all possible recording times for a signal of $14$ when the original domain $0\leq t\leq10$ is used, and explain the difference from part (b).

[2]

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Question 21

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

Two functions are defined for $0\leq x\leq8$ by

f(x)=\ln(x+2),\qquad g(x)=0.4x

The sum and difference functions are $s(x)=f(x)+g(x)$ and $d(x)=f(x)-g(x)$ .

I.

Write down expressions for $s(x)$ and $d(x)$ .

[2]

II.

State the domain of each of the two new functions.

[2]

Use your GDC to find the maximum value of $d$ and the value of $x$ at which it occurs.

[2]

Find the point, other than any endpoint, at which the graphs of $f$ and $g$ intersect.

[2]

Question 22

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

During a laboratory trial, a response index is modelled by

r(t)=te^{-0.4t},\qquad 0\leq t\leq12

where $t$ is measured in hours.

I.

Use your GDC to find the coordinates of the maximum point of the graph of $r$ .

[2]

II.

State the range of $r$ on the stated domain.

[2]

Find the two times at which the response index is $0.250$ .

[2]

Hence determine the length of time for which the response index exceeds $0.250$ . If you did not obtain the times in part (b), use $t=0.280$ and $t=8.94$ .

[2]

Question 23

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

The function $q$ is defined by

q(x)=x^4-8x^2+5

for $-3\leq x\leq3$ .

I.

Use your GDC to find all the zeros of $q$ on the stated domain.

[2]

II.

Find the coordinates of all turning points of $q$ on the stated domain.

[2]

Explain why the graph of $q$ is symmetric about the $y$ -axis.

[2]

Sketch the graph of $q$ on the stated domain, labelling the turning points and endpoint values.

[2]

Question 24

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

The functions $f$ and $g$ are defined by

f(x)=\frac{3x+1}{x-2},\qquad g(x)=0.5x+4

I.

State the equations of the vertical and horizontal asymptotes of the graph of $f$ .

[2]

II.

Find the coordinates of the intercepts of the graph of $f$ with the coordinate axes.

[2]

Use your GDC to find the points of intersection of the graphs of $f$ and $g$ .

[2]

Explain why a GDC window showing only $-4\leq x\leq4$ would lead to an incomplete conclusion about the intersections.

[2]

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Question 25

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

The vertical displacement of a floating sensor is modelled for $0\leq t\leq12$ by

h(t)=5+3\cos\left(\frac{\pi t}{6}\right)+0.2t

where $h(t)$ is measured in metres and $t$ in hours.

I.

Calculate $h(0)$ and $h(12)$ .

[2]

II.

Use your GDC to find the local maximum and local minimum points in the interior of the stated domain.

[2]

State the absolute maximum value of $h$ on the stated domain and explain why it is not the value at the local maximum.

[2]

Find the range of $h$ on the stated domain.

[2]

Question 26

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A sensor converts a concentration $c$ into a voltage using $f(c)=0.8c+1$ . A processor converts a voltage $v$ into an index using $g(v)=50\ln v$ . The concentration satisfies $0\leq c\leq10$ , and the overall index is $H(c)=(g\circ f)(c)$ .

I.

Find an expression for $H(c)$ .

[2]

II.

State the domain and range of $H$ in this context.

[2]

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

[2]

An index of $80$ is recorded. Determine the corresponding concentration and verify your result using composition.

[2]

Question 27

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The function $q$ is defined by

q(x)=2(x-1)^2-5

with domain $x\geq1$ . The function $r$ is defined by $r(x)=3x-2$ .

I.

Explain why the stated domain allows $q$ to have an inverse.

[2]

II.

Find $q^{-1}(x)$ and state its domain.

[2]

Find an expression for $(q^{-1}\circ r)(x)$ and state its domain.

[2]

Solve $(q^{-1}\circ r)(x)=5$ .

[2]

Question 28

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The function $p$ is defined by

p(x)=\frac{5x+2}{x+3}

with domain $x>-3$ .

I.

State the range of $p$ .

[2]

II.

Find $p^{-1}(x)$ and state its domain.

[2]

Verify algebraically that $(p^{-1}\circ p)(x)=x$ for $x>-3$ .

[2]

Calculate $p^{-1}(4.2)$ and interpret the result as a solution of an equation involving $p$ .

[2]

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Question 29

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A water-quality instrument measures pollutant concentration $x$ , in milligrams per litre, where $0\leq x\leq40$ . Its sensor voltage is modelled by

$s(x)=0.8\sqrt{x+1}+0.2$

A processor converts a voltage $v$ into a displayed reading using

$p(v)=120\ln v$

I.

Determine the range of $s$ in this context.

[2]

II.

Find an expression for the displayed reading $R(x)=(p\circ s)(x)$ and state its contextual domain.

[2]

I.

Calculate the displayed reading when $x=15$ .

[2]

II.

Find $R^{-1}(r)$ , expressing the pollutant concentration in terms of a displayed reading $r$ .

[2]

A displayed reading of $200$ is observed. Determine whether this reading is consistent with the stated concentration interval. Justify your answer.

[2]

Question 30

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A digital image stores a brightness value $x$ in the interval $-1\leq x\leq1$ . A normalization function and a contrast function are defined by

$f(x)=\frac{x+1}{2},\qquad g(u)=u^2$

where $0\leq u\leq1$ . A final function $h(v)=255v$ converts the result to a display level.

I.

Find an expression for $D(x)=(h\circ g\circ f)(x)$ .

[2]

II.

State the domain and range of $D$ in this context.

[2]

I.

Find $D^{-1}(y)$ for $0\leq y\leq255$ .

[2]

II.

A pixel has display level $180$ . Determine its original brightness value.

[2]

If contrast were applied before normalization, the intermediate output would be $(f\circ g)(x)$ . Since $g$ is defined for inputs in $[0,1]$ , compare the two orders on their common domain $x\in[0,1]$ . Show that the two orders give the same intermediate output only when $x=1$ .

[2]

Question 31

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

On the interval $-1\leq x\leq8$ , two functions are defined by

$f(x)=\sqrt{x+1},\qquad g(x)=x-1$

The sum and difference functions are $s=f+g$ and $d=f-g$ .

I.

Write down expressions for $s(x)$ and $d(x)$ .

[2]

II.

State the domain common to $s$ and $d$ .

[1]

Find the point at which the graphs of $f$ and $g$ intersect.

[2]

I.

Determine the coordinates of the maximum point of $d$ on the stated interval.

[2]

II.

Sketch the graph of $d$ , labelling its endpoints, maximum point and zero.

[3]

III.

Hence, state the range of $d$ on the stated interval.

[1]

Question 32

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The vertical position of a robotic arm is modelled by

h(t)=(t-6)^2+4

where $0\leq t\leq12$ and $t$ is measured in seconds. A camera records the transformed value

$R(t)=\ln(h(t))$

I.

State the range of $h$ .

[1]

II.

Explain why $h$ does not have an inverse function on $0\leq t\leq12$ .

[2]

I.

The arm is observed only during its return motion, $6\leq t\leq12$ . Find the inverse of the restricted function $h$ .

[2]

II.

State the domain of this inverse.

[1]

During the return motion, the camera records $R(t)=\ln13$ . Determine $t$ .

[2]

Find the inverse of $R$ during the return motion and state its domain.

[2]

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Question 33

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

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A medical device uses a patient's mass $w$ , in kilograms, where $30\leq w\leq150$ . It first calculates a body-size index

$m(w)=\sqrt{\frac{w+10}{80}}$

and then calculates a dose, in milligrams, using

$d(u)=240u+20$

Dose function

I.

Find the composite dose function $D=d\circ m$ .

[2]

II.

Determine the range of $D$ in this context.

[2]

Inverse dose function

I.

Let $r$ denote the displayed dose in milligrams. Find $D^{-1}(r)$ , including its domain and the units of the output.

[2]

II.

A dose of $300$ milligrams is displayed. Calculate the corresponding patient mass.

[2]

Explain why the composition $m\circ d$ does not represent the operation of this medical device.

[2]

Question 34

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A traveller exchanges an amount $x$ using Bank A. After its fee, the amount received is modelled by

$f(x)=0.92x-4$

The traveller later exchanges the full received amount back using Bank B, which applies

$g(y)=1.06y-3$

All monetary amounts are measured in the same reference currency for comparison.

I.

Find the round-trip function $T=g\circ f$ .

[2]

II.

Calculate the amount returned from an original amount of $500$ .

[2]

Find an expression for the round-trip loss $L(x)=x-T(x)$ , and calculate the loss when $x=500$ .

[2]

I.

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

[2]

II.

Explain why $g$ is not the inverse of $f$ .

[2]

Question 35

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A continuous, strictly increasing function $f$ has domain $[1,7]$ and range $[-2,5]$ . Its graph passes through $(1,-2)$ , $(3,0)$ , $(4,1)$ and $(7,5)$ .

On the same axes, sketch the graph of $f^{-1}$ . Label the images of the four given points.

[3]

I.

Write down $f^{-1}(1)$ .

[1]

II.

Write down $f^{-1}(5)$ .

[1]

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

[2]

Evaluate $(f^{-1}\circ f)(3)$ and $(f\circ f^{-1})(-2)$ .

[2]

The graph is extended by adding the point $(8,4)$ while remaining continuous. Explain why the extended function cannot have an inverse.

[1]

Question 36

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The functions $f$ and $g$ are defined by

f(x)=\sqrt{x-2},\qquad g(x)=\ln(7-x)

I.

Find an expression for $(f\circ g)(x)$ .

[2]

II.

Determine the domain of $f\circ g$ .

[2]

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

[2]

Solve $(f\circ g)(x)=1$ , giving your answer to three significant figures.

[2]

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Question 37

HL • Paper 2

Hard

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HL • Paper 2

Hard

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A calibration system applies two functions in sequence. The first is $A(x)=3x-7$ and the second is $B(y)=e^{y/20}$ . The overall calibration is $C=B\circ A$ .

I.

Find an expression for $C(x)$ .

[2]

II.

Find $A^{-1}(x)$ and $B^{-1}(x)$ .

[2]

Hence find $C^{-1}(x)$ , making clear the order in which the inverse functions are applied.

[2]

Determine the input that produces a calibrated output of $4.5$ , and verify the result.

[2]

Question 38

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The function $p$ is defined by

p(x)=x^3+x+1

for $x\in\mathbb R$ . The function $q$ is defined by $q(x)=2x-3$ .

I.

Use a graph to explain why $p$ has an inverse function.

[2]

II.

Use your GDC to find $p^{-1}(20)$ , giving your answer to three significant figures.

[2]

Without finding an algebraic formula for $p^{-1}$ , solve $(p^{-1}\circ q)(x)=2$ .

[2]

Explain why $(p^{-1}\circ p)(x)=x$ for every real $x$ .

[2]

Question 39

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The functions $f$ and $g$ are defined by

f(x)=4e^{0.3x}-2

for $x\in\mathbb R$ , and

g(x)=\sqrt{x+2}

for $x\geq-2$ .

I.

Find $f^{-1}(x)$ and state its domain.

[2]

II.

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

[2]

Find an expression for $(f^{-1}\circ g)(x)$ and state its domain.

[2]

Solve $(g\circ f)(x)=10$ .

[2]

Question 40

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $x\geq2$ , the function $f$ is defined by

$f(x)=x+\frac{4}{x}$

(a)

I.

Determine the minimum value and range of $f$ .

[2]

II.

Explain why $f$ has an inverse function on the stated domain.

[2]

(b)

I.

Find $f^{-1}(x)$ and state its domain.

[3]

II.

Verify that $(f^{-1}\circ f)(x)=x$ for $x\geq2$ .

[2]

The family $f_a(x)=x+\frac{a}{x}$ is restricted to $x\geq\sqrt a$ , where $a>0$ . Hence, write down $f_a^{-1}(x)$ and its domain.

[2]

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Question 41

HL • Paper 3

Hard

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HL • Paper 3

Hard

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For a real parameter $a$ , define

$f_a(x)=\frac{ax+1}{x-a},\qquad x\neq a$

I.

State the equations of the vertical and horizontal asymptotes of the graph of $f_a$ .

[2]

II.

State the domain and range of $f_a$ .

[2]

I.

Show that $f_a^{-1}(x)=f_a(x)$ .

[2]

II.

Hence, state the value of $(f_a\circ f_a)(x)$ for $x\neq a$ .

[2]

Find the coordinates of the two points where the graph of $f_a$ intersects the line $y=x$ .

[2]

The horizontal distance between these two intersection points is $4$ . Determine the possible values of $a$ .

[2]

Question 42

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The functions $f$ and $g$ are defined by

$f(x)=\sqrt{9-x},\qquad g(x)=x^2+1$

using their largest possible real domains.

The graph is shown over a finite plotting window; any curve endpoints visible within that window are plotting truncations, unless forced by the stated domain.

I.

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

[2]

II.

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

[2]

Explain why simplifying $(g\circ f)(x)$ to $10-x$ does not allow its domain to be changed to all real numbers.

[2]

I.

Find $f^{-1}(x)$ and state its domain.

[2]

II.

Verify that $(f\circ f^{-1})(x)=x$ on the domain of $f^{-1}$ .

[2]

Solve $(f\circ g)(x)=2$ on its domain.

[2]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The function $f$ is defined by

f(x)=|2x-6|+1

for all real $x$ .

I.

State the range of $f$ .

[1]

II.

Explain why $f$ has no inverse on $\mathbb R$ .

[2]

I.

Restrict the domain to $x\geq3$ . Find the inverse function $r$ .

[2]

II.

Restrict the domain instead to $x\leq3$ . Find the inverse function $l$ .

[2]

Determine $(r\circ f)(x)$ separately for $x\geq3$ and $x<3$ .

[2]

Explain geometrically why $(r\circ f)(x)=6-x$ when $x<3$ .

[2]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $k>0$ , define

$f_k(x)=\frac{k}{x+1},\qquad x\geq0$

$x$	$f_k(x)$
0	$k$
1	$\frac{k}{2}$
2	$\frac{k}{3}$

I.

State the range of $f_k$ and explain why $f_k$ has an inverse.

[2]

II.

Find $f_k^{-1}(x)$ and state its domain.

[2]

Find the positive fixed point of $f_k$ in terms of $k$ .

[2]

I.

Find an expression for $(f_k\circ f_k)(x)$ .

[1]

II.

Show that the non-negative solutions of $(f_k\circ f_k)(x)=x$ are precisely the fixed points of $f_k$ .

[2]

Determine $k$ if the fixed point is $x=2$ .

[2]

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Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

An unfamiliar function is defined by

$h(x)=x+e^{-x},\qquad x\in\mathbb R$

I.

Use graphing technology to determine the coordinates of the minimum point of $h$ .

[2]

II.

State the range of $h$ .

[2]

Explain why $h$ has no inverse function on $\mathbb R$ , and state a domain restriction that retains the point $(0,1)$ and makes $h$ one-to-one.

[2]

I.

Solve $h(x)=2$ on the restricted domain $x\geq0$ .

[1]

II.

Solve $h(x)=2$ on the restricted domain $x\leq0$ .

[1]

Deduce the number of real solutions of $h(x)=c$ for each of the cases $c<1$ , $c=1$ and $c>1$ .

[2]

Question 46

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The concentration of a medicine $t$ hours after administration is modelled by

f(t)=100e^{-0.2t},\qquad t\geq0

A response score is calculated from concentration $c$ using

g(c)=\frac{c}{c+20},\qquad c>0

The response as a function of time is $h=g\circ f$ .

I.

Find and simplify an expression for $h(t)$ .

[2]

II.

State the domain and range of $h$ in this context.

[2]

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

[2]

Determine when the response score first falls to $0.600$ .

[2]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The family of curves

$y=h_k(x)=x^3-3x+k$

is compared with the line $y=x$ , where $k$ is a real parameter. Intersections satisfy

$x^3-4x+k=0$

I.

For $k=0$ , find all points of intersection.

[2]

II.

For $k=2$ , determine the three points of intersection, giving coordinates to three significant figures.

[2]

Let $F_k(x)=x^3-4x+k$ . Determine the coordinates of the local maximum and local minimum of $F_k$ in terms of $k$ .

[4]

I.

Hence, determine the interval of values of $k$ for which the cubic and the line have three distinct intersections.

[2]

II.

Describe the intersections when $k=\frac{16}{3\sqrt3}$ .

[2]

Explain why changing the graphing window could lead to an incorrect conclusion about the number of intersections.

[2]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two reversible coding steps are represented by the affine functions

f(x)=1.5x+2, \qquad g(x)=0.8x-5

I.

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

[2]

II.

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

[2]

Explain why there is no value of $x$ for which the two coding orders give the same output.

[2]

Let $F(x)=ax+b$ and $G(x)=cx+d$ , where $a\neq0$ and $c\neq0$ . Show that $F$ and $G$ commute if and only if

$d(a-1)=b(c-1)$

[3]

I.

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

[2]

II.

State, in the correct order, the composition that reverses $f\circ g$ .

[1]

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