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Functions

UNIT 10 Functions62

UNIT 10

Functional relationshipsA cinema manager wants to study the relationship between the number of cinema-goers and the money obtained from ticket sales.

Number of cinema-goers 15 20 36 54

Ticket sales (€) 120 160 288 432

The value of the ticket sales depends on the number of cinema-goers. For a given number of cinema-goers, there is only one value for ticket sales. Also, we can choose a value for the number of cinema-goers and get the corresponding value for the ticket sales.

A function is a relation between two variables, x and y, in which each value of x corresponds to a single value of y.

The variable whose value can be freely chosen is called the independent variable, and is given the letter x.

The variable whose value is determined by the functional relationship is the dependent variable, and is given the letter y.

Worked example 1 Decide if the following relations are functions.

a) each real number, x, and its double, yb) the length of time a tap is running and how much water comes outc) the weights and heights of a group of people

Solutiona) It’s a function. For each value of x, there's only one value of y. b) It’s a function because the amount of water that comes out is

dependent on the length of time the tap is running.c) It isn’t a function, because people with the same height may have

different weights.

Ways of expressing a function Statement: this is the description of the functional relationship.

Álvaro travels 600 m at a constant speed for 2 minutes.

Table of values: this is the set of related values.

Time (s) 20 40 60 80 100 120

Distance (m) 100 200 300 400 500 600

Graph: this represents the points that relate to the function. The independent variable, x, is shown on the X-axis, and the dependent variable, y, on the Y-axis.

Algebraic equation: also called the functional equation, this shows how the y values are related to the x values.

The equation for this graph is y = 5x.

1

Lost in translationAxis labels in Spanish are written using capital italic letters (X-axis, Y-axis). However, in English they’re written with lower case italic letters (x-axis, y-axis).

Mathematical language The dependent variable y can

also be called the image of x.

We can use f(x) to represent the variable y. This indicates that y is dependent on x.

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Activities

UNIT 10 Functions 63

11 Listen and decide if these statements about functions are true or false. Review and explain your answers with a classmate.

Each value of x corresponds/doesn’t correspond to a single value of y.

So it is/isn't a function.

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2 Decide whether or not each graph represents a function. Explain your answers in your notebook.

a)

O X

Y c)

O X

Y

b)

O X

Y d)

O X

Y

3 We know that 1 kg of oranges costs €1,20. a) Make a table of values for different amounts of

oranges and say which are the independent and dependent variables.

b) Does it make sense to give x negative values? c) Does it make sense to give x values that are not

whole numbers?

4 Decide which of these are functions, and if so, identify the independent and dependent variables.

a) the price for a given number of kilograms of pears b) the decimal equivalents of each fraction c) each person and their age d) each number and its half

5 Given that the ski lift on a ski slope runs at a constant 4 m/s, copy and complete the table in your notebook.

Time (s) 5 15 50

Distance (m) 500 800 2 000

6 Find the image of x = 5 and x = −2 using these functions.

a) f(x) = 2x - 1 b) f(x) = -2x - 1

7 Write the algebraic equation for: a) the function that relates each number to its

triple, plus 1. b) the function that relates each number to its half. c) the function that relates each number to its

negative.

8 This table shows the time taken for a certain number of students to decorate a room for a party.

Time (h) 6 3 2 1

Number of students 1 2 3 6

If the job had to be done in half an hour, how many students would be needed?

9 Find an algebraic equation for: a) the function that gives the perimeter of an

equilateral triangle based on its side lengths. b) the function that relates the radius of a circle’s

circumference to its length.

10 Decide whether or not each table represents a function, and write, where possible, an algebraic equation.

a) x 1 2 3 4

y 3 5 7 9

c) x 1 1 1 1

y 1 4 9 16

b) x 0 1 2 3

y -2 -1 0 1

d) x 2 4 6 8

y 1 2 3 4

Take noteThe image of x in a function is the value we obtain when solving f(x) for a given value of x.

UNIT 10 Functions

Domain, range and intercepts Domain and rangeSandra’s father explains how long to bake a cake: first we have to wait for the oven to reach 180 ºC. Then we have to put the cake mix in the oven and leave it to cook for 30 minutes.

Sandra has made a graph showing the temperature of the oven versus time.

In the first 10 minutes after turning on the oven, the temperature rises from 20 ºC, the room temperature, to 180 ºC, the desired temperature. From minute 10 to 40, it maintains that temperature. When the oven is switched off, the temperature drops until it reaches room temperature; this takes 20 minutes.

The variation in time was from 0 to 60 minutes, so the domain is the interval [0, 60].

The temperature varies between 20 ºC and 180 ºC, so the range is the interval [20, 180].

The domain of a function is the set of all possible values of the independent variable, called Dom f. The range of a function is the set of all possible resulting values of the

dependent variable.

Worked example 12 Find the domain of these functions using algebra.

a) f (x ) = x2 b) g(x ) = x− 1 c) h(x ) =1

x−2

Solutiona) For any real number x, we obtain an image by squaring it; therefore,

the domain is all real numbers: Dom f = Rb) We can find an image only for values that make the number under

the radical sign positive, that is to say when x ³ 1. So, the domain is real numbers greater than or equal to 1. We write this as:

Dom g = [1, +¥)c) We can find an image as long as the denominator is not 0, which

happens when x ≠ 2. So, the domain is all real numbers except 2: Dom h = R - {2}

Axis interceptsThe axis intercepts are any points where the graph crosses the axes.

The Y-axis intercepts take the form (x, 0), where the value of x is found by solving f(x) = 0. The X-axis intercepts take the form (0, y), where the value of y is

found by solving f(0).

2

Take noteA function can cross the X-axis at various points, but can cross the Y-axis only once.

O 1

1

X

Y

X-axis intercepts:

(-2, 0) and (2, 0)

Y-axis intercept: (0, -4)

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Activities

65UNIT 10 Functions

13 In your notebook, write the domain and range. a)

O 1

1

Y

X

c)

O 1

1

X

Y

b)

O 1

1

X

Y d)

O 1

1

X

Y

14 Find the domain, range and intercepts. a)

O 1

1

X

Y d)

O 1

1

Y

X

b)

O 1

1

X

Y e)

O 1

1

X

Y

c)

O 1

1

X

Y f)

O 1

1

Y

X

15 Find the domain of each function using algebra. a) f (x ) = x2- 5x + 6 d) f (x ) = x

b) f (x ) =1x

e) f (x ) =x

x + 2

c) f (x ) = x2 + x f) f (x ) =x + 1x−5

16 The area of a rectangle is 18 cm2.

a) Use algebra to write the relation between the base variable and the height of the rectangle.

b) Find the domain of the function produced by the formula you wrote for part a).

17 Find the axis intercepts of these functions and use them to draw graphs of the functions.

a) y = −x c) y = x + 5 b) y = −x− 1 d) y = x−5

Worked example 18 Find the axis intercepts for f(x) = x2 - 2x - 3.

Solution X-axis intercepts: If f(x) = 0 → x2 - 2x - 3 = 0

f(−x)=(−x)2−(−x)=x2+x≠f(x)−f(x)=−(x2−x)=−x2+x≠−f(x)

⎫⎬⎪⎪

⎭⎪⎪

→ x =−(−2) ± 16

2 ⋅ 1=

2 ± 4

2→ x = −1

x = 3

⎧⎨⎪⎪⎪

⎩⎪⎪⎪

→ x =−(−2) ± 16

2 ⋅ 1=

2 ± 4

2→ x = −1

x = 3

⎧⎨⎪⎪⎪

⎩⎪⎪⎪

→ x =−(−2) ± 16

2 ⋅ 1=

2 ± 4

2→ x = −1

x = 3

⎧⎨⎪⎪⎪

⎩⎪⎪⎪

The X-axis intercepts are: (-1, 0) and (3, 0) Y-axis intercepts: If x = 0 → f(0) = -3 The function intercepts the Y-axis at: (0, -3)

19 Find the axis intercepts of these functions. a) y = 2x2 - 2 b) y = x2 - 1 c) y = -x2 + 3x

20 Work with a classmate. Uses the phrases below to solve this problem. A game uses €1 coins. With one you can play for 30 minutes, and with each additional coin, 60 minutes, to a

maximum of 180 minutes. Draw the function’s graph. How much does it cost to play for 20, 80 and 120 minutes? The independent/dependent variable is the time/number of coins. It costs ... to play for ... minutes.

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65

UNIT 10 Functions

ContinuityThis graph shows Pablo’s height, in cm, between the ages of 6 and 16 years old.

O 6 7 8 9 10 11 12 13 14 15 16

110

140

170

Age

Height (cm)

The graph can be drawn without lifting your pencil. This means that Pablo’s height on the given interval is a continuous function.

A function is continuous on an interval when its graph doesn't have any jumps or interruptions.

Not all functions are continuous within their domains. Any point where the function shows a jump is called a point of discontinuity.

Increasing and decreasing functionsGermán goes for a bike ride. This graph shows the results of his pulse meter.

O 1

60

70

80

90

2 3 4 5 Distance (km)

Heart rate(BPM)

In the first kilometre, Germán’s number of beats per minute (BPM) increased from 60 to 90. It then dropped to 75 during a 2 km downhill stretch. After that the terrain was flat and Germán’s heart rate stayed at 75.

We can say that this function was increasing on the interval (0, 1), decreasing on the interval (1, 3) and constant on the interval (3, 5).

A function is increasing on an interval if the dependent variable f(x) increases as the independent variable x increases.

A function is decreasing on an interval if the dependent variable f(x) decreases as the independent variable x increases.

A function is constant on an interval if it neither increases nor decreases.

A turning point (a, f(a)) of a continuous function is a maximum point where the function changes from increasing to decreasing.

A turning point (a, f(a)) of a continuous function is a minimum point where the function changes from decreasing to increasing.

3

4

Take noteWhen we draw a graph, we can adjust the scales of the axes or mark an interval to produce a clearer representation of the function.

Take noteTo show the points of discontinuity in a function, we use the formula x = a, where a is the point on the X-axis where the function is not continuous.

Mathematical language On a graph, we use a solid

dot • to show that the value of y is a value of the function for the given value of x. We use an open dot O when the y value isn’t a value of the function for the given value of x.

To find the relative extrema of a function means to show its maximum and minimum points.

Evaluating the monotonicity of a function is determining if it is increasing or decreasing in its domain, and finding its relative extrema (maximum and minimum points).

66

Activities

67UNIT 10 Functions

Worked example 21 Decide which of these functions are continuous

and explain your answer. Evaluate the monotonicity of each function.

a)

O 1

1

X

Y c)

1 XO

1

Y

b)

1 XO

1

Y d)

O 1

1

X

Y

SolutionFunctions b) and c) have jumps at x = 2, so they aren’t continuous. Functions a) and d) don’t have jumps, so they’re continuous.Monotonicitya) It's increasing on the intervals (-¥, 2) and

(4, +¥) and decreasing on (1, 4). Its maximum point is (2, 4). Its minimum point is (4, 0).

b) It's decreasing on (-¥, 2) and (2, +¥). It doesn’t have relative extrema.

c) It's increasing on (0, 2) and decreasing on (-¥, 0) and (2, +¥). Its minimum point is (0, -2).

d) It's increasing on the intervals (-¥, 1) and (3, +¥) and decreasing on the interval (1, 3). Its maximum point is (1, 3). Its minimum point is (3, -1).

22 Do these statements describe continuous functions? a) a quantity of chocolates and how much they cost b) the growth of a tree and the time since it was planted

23 Decide if these functions are continuous. If any aren’t, write their points of discontinuity.

a)

O 1

1

X

Y

b)

O 1

1

X

Y

24 Evaluate the monotonicity of these functions. a)

O 1

1

X

Y

b)

O 1

1

X

Y

25 Create a table of values on the interval [−4, 4] for the function f (x ) =

1x

and then draw the graph. Is

the interval continuous?

26 Copy the table in your notebook. Listen and complete the number of newspapers sold by the kiosk each day. Then work with a classmate to describe the increase and decrease in sales during the week.

Day M T W Th F S Su

Sales

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67

UNIT 10 Functions

Symmetry and periodicitySymmetryLook at these functions.

O 1

1

X

Y

O 1

1

X

Y

x -2 -1 0 1 2

f(x) 4 1 0 1 4

x -2 -1 0 1 2

f(x) -8 -1 0 1 8

For every point (a, b) on the graph, there’s also a point (-a, b):

For every point (a, b) on the graph, there’s also a point (-a, -b):

f(1) = 1 = f(-1)

f(2) = 4 = f(-2)

f(1) = 1 = -f(-1)

f(2) = 8 = -f(-2)

This is an even function. This is an odd function.

A function has even symmetry if its graph is symmetric with respect to the Y-axis.

f(−x) = f(x)

A function has odd symmetry if its graph is symmetric with respect to the origin.

f(−x) = −f(x)

PeriodicityA scooter factory is planning a new model with an integrated battery and a motor that allows it to move using an accelerator.

The manufacturing process ends when a finished scooter arrives at the warehouse. The whole process takes 15 minutes.

This graph represents how the factory operates.

This function is periodic because it behaves in the same way same across equal intervals of 15 minutes. Here, 15 is called the period.

A function is periodic for period T when the behaviour of the function in an interval [x, x + T ] is repeated in successive intervals.

5

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68

Activities

69UNIT 10 Functions

33 Work with a classmate to decide whether or not these phenomena are periodic functions. If you have enough information, say what the period is.

a) the height of a rubber ball that bounces off the floor after it is dropped b) the height above ground of a Ferris wheel car in relation to time passed

CLIL zone

27 What type of symmetry do these functions have? a)

O 1

1

X

Y

b)

O 1

1

X

Y

c)

O 1

1

X

Y

d)

O 1

1

X

Y

28 What type of symmetry do these functions have? a) x -2 -1 0 1 2

f(x) 16 1 0 1 16

b) x -2 -1 0 1 2

f(x) -32 -1 0 1 32

Worked example 29 Check the symmetry of these functions. a) f(x) = x2 c) f(x) = x3 - x b) f(x) = 2x4 - x2 d) f(x) = x2 - x

Solution a) f(-x) = (-x)2 = x2 = f(x) The function is even. b) f(-x) = 2(-x)4 - (-x)2 = 2x4 - x2 = f(x) The function is even. c) f (−x ) = (−x )3 − (−x ) = −x3 + x

−f (x ) = −(x3 − x ) = −x3 + x

⎫⎬⎪⎪⎪

⎭⎪⎪⎪

f (−x ) = −f (−x )

f (−x ) = (−x )3 − (−x ) = −x3 + x

−f (x ) = −(x3 − x ) = −x3 + x

⎫⎬⎪⎪⎪

⎭⎪⎪⎪

f (−x ) = −f (−x )

f (−x ) = (−x )3 − (−x ) = −x3 + x

−f (x ) = −(x3 − x ) = −x3 + x

⎫⎬⎪⎪⎪

⎭⎪⎪⎪

f (−x ) = −f (−x )

f (−x ) = (−x )2 − (−x ) = x2 + x ≠ f (x )−f (x ) = −(x2 − x ) = −x2 + x ≠ −f (x )

⎫⎬⎪⎪

⎭⎪⎪

→The function is odd.

d) f (-x ) = (-x )2 - (-x ) = x2 + x ≠ f (x )-f (x ) = -(x2 - x ) = -x2 + x ≠ -f (x )

The function neither even nor odd.

30 For each function, say if it’s odd, even or without symmetry.

a) f (x ) = −x b) f (x ) =1

x2 c) f (x ) = x5 − x

31 Find the period for each of these periodic functions. a)

O 1

1

X

Y b)

O

1

1 X

Y

32 Ana has accidentally deleted the graph from x = 3 on. It’s a periodic function with a period of T = 5. Draw it in your notebook.

70 UNIT 10 Functions

Interpreting graphsSome friends are spending a weekend at a campsite. There’s a water tank for taking showers.

The friends get up at 6 a.m. but can’t shower yet because the tank is empty. They decide to make breakfast while the water tank fills up.

From 7 to 8 a.m. they don’t use the showers, then from 8 to 9 a.m. they use 4,5 m3 of water. They go shopping from 9 to 10 a.m. From 10 to 11 a.m. they use the remaining water for washing dishes.

This graph plots time versus the amount of water in the tank, and shows the state of the water tank from 6 a.m. to 11 a.m.

To interpret the graph, look at the function’s characteristics.

Domain: [6, 11]Range: [0, 5]X-axis intercepts: (6, 0) and (11, 0)It’s continuous.Increasing on (6, 7), decreasing on (8, 9) and (10, 11) and constant on (7, 8) and (9, 10). No maximum or minimum points. Neither symmetric nor periodic.

We follow these steps to interpret a graph of a function.

1. Identify the independent and dependent variables of the function.

2. Find the domain and the range of the function.

3. Find the axis intercepts.

4. Decide if it’s continuous.

5. Find the increasing and decreasing intervals.

6. Find any maximum and minimum points.

7. Determine its symmetry.

8. Decide whether or not the function is periodic.

6

Worked example 34 Look at the graph and describe all the function’s features.

Solution

Domain: [0, +∞)[0, +∞) Range: [−4, +∞)[−4, +∞) X-axis intercepts: (0, 0), (1, 0) and (3, 0) Y-axis intercepts: (0, 0)Continuity: it isn’t continuous. It has a point of discontinuity at (3, 1).Monotonicity: It’s increasing on (0; 0,5) Maximum: (0,5; 1) and on (2, +∞)(2, +∞) It’s decreasing on (0,5; 2) Minimum: (2, -4)Symmetry: it’s neither even nor odd. Periodicity: it isn’t periodic.

O 1

1

X

Y

Daily LifeThe graph of a function can help us visualise the aspects of a real-life situation.

Take noteTo interpret a function graph, read it from left to right.

O 6

1

X

Y

7 8 9 10 11

2

3

4

5

12

Activities

71UNIT 10 Functions

38 This graph shows the changes in altitude (m) of a motorbike race over the course of 50 km. Listen to the students describing the graph and identify three mistakes they make.

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35 Miguel, Carlos, Ana and Julia participated in a race. In your notebook, match each graph with its competitor.

I Miguel started slowly and progressively increased his speed.

II Carlos started fast and gradually slowed down. III Ana went slowly for the first half of the race and

faster in the second half. IV Julia kept a steady pace throughout the race. a)

O 1

5

X

Y

b)

O 1

5

X

Y

c)

O 1

5

X

Y

d)

O 1

5

X

Y

36 This graph shows the sales of electric bikes and mountain bikes in a shop during the last six years.

O

20

1 X

Electric bikes

Y Mountain bikes

a) How many bikes of each type did the shop sell in the first year?

b) Which year had the biggest difference between the sales of electric bikes and mountain bikes?

c) Did the sales coincide at any point?

37 This graph shows the price, in euros, of 1 kg of hake over one year, from January to December.

J D

9

12

15

18

X

Y

a) Between what values did the price per kg vary? b) In which month did the price rise the most? c) For how long did the price stay constant? d) In which month did it reach its maximum value? e) How long was the longest uninterrupted price rise? f) Find the periods in which there was a reduction in

price, and what each was worth to the consumer.

Y

O 10

200

X