Functions: Characteristics and Properties - High School Math Exercises, Study Guides, Projects, Research of Mathematics

Download Functions: Characteristics and Properties - High School Math Exercises and more Study Guides, Projects, Research Mathematics in PDF only on Docsity!

x NEL

GOALS

You will be able to

• Review and consolidate your knowledge of the properties and characteristics of functions and their inverses
• Review and consolidate your knowledge of graphing functions using transformations
• Investigate the characteristics of piecewise functions

What type of function can be used to model the height of a golf ball during its flight, and what information about the relationship between height and time can be found using this function?

Chapter

?

Functions:

Characteristics

and Properties

NEL 1

NEL Chapter 1 3

APPLYING What You Know

Modelling the Height of a Football

During a football game, a football is thrown by a quarterback who is 2 m tall. The football travels through the air for 4 s before it is caught by the wide receiver.

What function can be used to model the height of the football above the ground over time?

A. Explain why the variables time, t , in seconds and height, in metres are good choices to model this situation.

B. What is What does it mean in the context of this situation?

C. What happens at

D. What happens at

E. Explain why each of the following functions is not a good model for this situation. Support your claim with reasons and a well-labelled sketch. i) ii) iii)

F. Determine a model that can be used to represent the height of the football, given this additional information:

- The ball reached a maximum height of 22 m. - The wide receiver who caught the ball is also 2 m tall.

G. Use your model from part F to graph the height of the football over the duration of its flight.

h ( t ) 5 5 t^2 1 4 t 2 3

h ( t ) 5 2 5 ( t 2 4 ) 2 1 2

h ( t ) 5 2 5 t ( t 2 4 )

t 5 4 s?

t 5 2 s?

h ( 0 )?

h ( t ),

?

Getting Started

4 1.1^ Functions NEL

LEARN ABOUT the Math

Jonathan and Tina are building an outdoor skating rink. They have enough materials to make a rectangular rink with an area of about and they do not want to purchase any additional materials. They know, from past experience, that a good rink must be approximately 30 m longer than it is wide.

1800 m^2 ,

Functions

GOAL Represent and describe functions and their characteristics.

? What dimensions should they use to make their rink?

EXAMPLE 1 Representing a situation using a mathematical model

Determine the dimensions that Jonathan and Tina should use to make their rink. Solution A: Using an algebraic model Let x represent the length. Let y represent the width.

The width, in terms of x , is Let represent the difference between the length and the width.

where

x 2

1800 x^5

f ( x ) 5 30.

f ( x ) 5 x 2 1800 x ,

f ( x )

1800 x.

1800 x^5 y

1800 5 xy

A 5 xy We know the area must be 1800 m^2 , so if we let the width be the independent variable , we can write an expression for the length.

Using function notation , write an equation for the difference in length and width. The relation is a function because each input produces a unique output. In this case the difference or value of the function must be 30.

YOU WILL NEED

• graph paper
• graphing calculator (optional)

6 1.1^ Functions NELNEL

The dimensions that are 30 m apart and produce an area of are

A function can also be represented with a graph. A graph provides a visual display of how the variables in the function are related.

Solution C: Using a graphical model Let x represent the length. Let y represent the width.

The width, in terms of x , is

Let represent the difference between the dimensions.

Determine the appropriate window settings to graph f ( x )on a graphing calculator.

f ( x ) 5 x (^2 1800) x

f ( x )

1800 x.

1800 x^5 y

1800 5 xy

A 5 xy

60 m 3 30 m.

1800 m^2

Create a table of values to investigate the difference between the length and the width for a variety of lengths.

Using length ( x ) as the independent variable, write an expression for width ( y ).

The value for x (length of rink) will be positive but surely less than 75 m, so we use and. We use the same settings for the range of f( x ), for simplicity.

Xmin 5 0 Xmax 5 75

Area (m^2 ) Length (m) Width (m) Length – Width 1800 100 18 82 1800 90 20 70 1800 80 22.5 57. 1800 70 25.71 44. 1800 60 30 30 1800 50 36 14 1800 40 45 25 (^1800 30 60 ) (^1800 20 90 )

APPLY the Math

Use the TRACE feature on the graph to investigate points with the ordered pairs (length, length width) on f ( x ).

2

To determine the length that is 30 m longer than the width, graph in and locate the point of intersection for and f ( x ).

g ( x )

g ( x ) 5 30 Y 2

For help using the graphing calculator to find points of intersection, see Technical Appendix, T-12.

Tech (^) Support

EXAMPLE 2 Using reasoning to decide whether a relation is a function

Decide whether each of the following relations is a function. State the domain and range.

a) b) c)

• • 0 1 2

0 1 2 3

y 5

y x 2

x

4 2

0

–4 –2 2 4

NEL Chapter 1 7

Graph the difference function.

A length of 50 m gives a 14 m difference between the length and the width.

Determine the length that exceeds the width by 30 m.

The dimensions that are 30 m apart and produce an area of are

Reflecting

A. Would the function change if width was used as the independent variable instead of length? Explain.

B. Is it necessary to restrict the domain and range in this problem? Explain.

C. Why was it useful to think of the relationship between the length and the width as a function to solve this problem?

60 m 3 30 m.

1800 m^2

Solution

or

R 5 5 h ( t ) P R 0 1 # h ( t ) # 76

D 5 5 t P R 0 0 # t # 900 6

h ( t ) 5 a cos 3 k ( t 2 d ) 4 1 c

h ( t ) 5 a sin 3 k ( t 2 d ) 4 1 c (^) This situation involves circular motion,

which can be modelled by a sine or cosine function.

Examine the conditions on the independent variable time to determine the domain. Time cannot be negative, so the lower boundary is 0. The wheel rotates once every 90 s, and Naill rides for 10 complete rotations.

The upper boundary is 900 s.

90 3 10 5 900

Examine the conditions on the dependent variable height to determine the range. The radius of the wheel is 3 m. Since the axle is located 4 m above the ground, the lowest height that Naill can be above the ground is the difference between the height of the axle and the radius of the wheel: This is the lower boundary of the range. The greatest height he reaches is the sum of the height of the axle and the radius of the wheel: This is the upper boundary of the range.

4 1 3 5 7 m.

4 2 3 5 1 m.

3 m 6

6

3 m

1 m

EXAMPLE 3 Using reasoning to determine the domain and range of a function

Naill rides a Ferris wheel that has a diameter of 6 m. The axle of the Ferris wheel is 4 m above the ground. The Ferris wheel takes 90 s to make one complete rotation, and Naill rides for 10 rotations. What are the domain and range of the function that models Naill’s height above the ground, in terms of time, while he rides the Ferris wheel?

NEL Chapter 1 9

10 1.1^ Functions NEL

In Summary

Key Ideas

• A function is a relation in which there is a unique output for each input. This means that each value of the independent variable (the domain) must correspond to one, and only one, value of the dependent variable (the range).
• Functions can be represented graphically, numerically, or algebraically.

Need to Know

• Function notation, is used to represent the values of the dependent variable in a function, so
• You can use the vertical line test to check whether a graph represents a function. A graph represents a function if every vertical line intersects the graph in, at most, one point. This shows that there is only one element in the range for each element in the domain.
• The domain and range of a function depend on the type of function.
• The domain and range of a function that models a particular situation may need to be restricted, based on the situation. For example, negative values may not have meaning when dealing with variables such as time.

y 5 f ( x ).

f ( x ),

Graphical Example Numerical Examples Algebraic Examples

y 4

0

2

–4 –2 2 4

x

Set of ordered pairs:

Table of values:

Mapping diagram:

• 1 3 5

0 2

5 (1, 3), (3, 5), ( 2 2, 9 ), (5, 11) 6 or f ( x ) 5 2 sin ( 3 x ) 1 4

y 5 2 sin ( 3 x ) 1 4

x y 22 4 21 1 0 0 1 1 2 4

y

0

x

function

y

0

x

not a function

8. Consider what happens to a relation when the coordinates of all its ordered pairs are switched. a) Give an example of a function that is still a function when its coordinates are switched. b) Give an example of a function that is no longer a function when its coordinates are switched. c) Give an example of a relation that is not a function, but becomes a function when its coordinates are switched. 9. Explain why a relation that fails the vertical line test is not a function. 10. Consider the relation between x and y that consists of all points such that the distance from to the origin is 5. a) Is (4, 3) in the relation? Explain. b) Is (1, 5) in the relation? Explain. c) Is the relation a function? Explain. 11. The table below lists all the ordered pairs that belong to the function g ( x ).

( x , y )

( x , y )

x 0 1 2 3 4 5 g ( x ) 3 4 7 12 19 28

a) Determine an equation for b) Does? Explain.

12. The factors of 4 are 1, 2, and 4. The sum of the factors is The sum of the factors is called the sigma function. Therefore, a) Find and c) Is? b) Is? d) Are there others that will work? 13. Make a concept map to show what you have learned about functions. Put “FUNCTION” in the centre of your concept map, and include the following words:

f ( 15 ) 5 f ( 3 ) 3 f ( 5 )

f ( 6 ), f ( 7 ), f ( 8 ). f ( 12 ) 5 f ( 3 ) 3 f ( 4 )

f ( 4 ) 5 7.

g ( 3 ) 2 g ( 2 ) 5 g ( 3 2 2 )

g ( x ).

algebraic model dependent variable domain function notation

graphical model independent variable mapping model

numerical model range vertical line test

Extending

14. Consider the relations and Draw the graphs of these relations, and determine whether each relation is a function. State the domain and range of each relation. 15. You already know that y is a function of x if and only if the graph passes the vertical line test. When is x a function of y? Explain.

x^2 1 y^2 5 25 y 5 " 25 2 x^2.

T

C A concept map is a type of web diagram used for exploring knowledge and gathering and sharing information. A concept map consists of cells that contain a concept, item, or question and links. The links are labelled and denote direction with an arrow symbol. The labelled links explain the relationship between the cells. The arrow describes the direction of the relationship and reads like a sentence.

Communication (^) Tip

NEL Chapter 1 13

14 1.2^ Exploring Absolute Value NEL

1.2 (^) Exploring Absolute Value

EXPLORE the Math

An average person’s blood pressure is dependent on their age and gender. For example, the average systolic blood pressure, for a 17-year-old girl is about 127 mm Hg. (The symbol mm Hg stands for millimetres of mercury, which is a unit of measure for blood pressure.) The average systolic blood pressure for a 17-year-old boy is about 134 mm Hg. When doctors measure blood pressure, they compare the blood pressure to the average blood pressure for people in the same age and gender group. This comparison, is calculated using the formula where P is the blood pressure reading and is the average reading for people in the same age and gender group.

How can the blood pressure readings of a group of people be compared?

A. Jim is a 17-year-old boy whose most recent blood pressure reading was 142 mm Hg. Calculate for Jim.

B. Joe is a 17-year-old boy whose most recent blood pressure reading was 126 mm Hg. Calculate for Joe.

C. Compare the values of and that were used to determine for each boy. What do you notice?

D. Complete the following table by calculating the values of for the given blood pressure readings for 17-year-old boys.

E. Draw a scatter plot of Pd as a function of blood pressure, P.

Pd

Pd

P 2 Pn 0 P 2 Pn 0

Pd

Pd

?

Pn

Pd , Pd 5 0 P 2 Pn 0 ,

Pn ,

YOU WILL NEED

• graph paper
• graphing calculator

GOAL Discover the properties of the absolute value function.

To use the absolute value command on a graphing calculator, press MATH and scroll right to NUM. Then press ENTER.

Tech (^) Support

Blood Pressure Reading, P^95 100 105 110 115 120 125 130 135 140 145 150 155

Pd

16 1.2^ Exploring Absolute Value NEL

FURTHER Your Understanding

1. Arrange these values in order, from least to greatest: 2. Evaluate. a) c) e)

b) d) f )

3. Express using absolute value notation. a) c) b) d) 4. Graph on a number line. a) b) c) d) 5. Rewrite using absolute value notation.

a)

b)

c)

d)

6. Graph and a) What do you notice? b) How could you have predicted this? 7. Graph the following functions. a) b) c) d) 8. Compare the graphs you drew in question 7. How could you use transformations to describe the graph of 9. Predict what the graph of will look like. Verify your prediction using graphing technology. 10. Predict what the graph of will look like. Verify your prediction using graphing technology.

f ( x ) 5 3 2 0 2 x 2 50

f ( x ) 5 0 2 x 1 10

f ( x ) 5 0 x 1 30 2 4?

f ( x ) 5 0 x 0 2 2

f ( x ) 5 0 x 1 20

f ( x ) 5 0 x 0 1 2

f ( x ) 5 0 x 2 20

f ( x ) 5 0 x 2 80 g ( x ) 5 0 2 x 1 80.

–4 –3–2 –1 0 1 2 3 4

–4 –3–2 –1 0 1 2 3 4

–4 –3–2 –1 0 1 2 3 4

–4 –3–2 –1 0 1 2 3 4

0 x 0 , 8 0 x 0 $ 16 0 x 0 # 2 4 0 x 0. 2 7

28 # x # 8 x 2 65

x , 2 3 or x. 3 x # 2 1 or x $ 1

To show that a number is not included in the solution set, use an open dot at this value. A solid dot shows that this value is included in the solution set.

Communication Tip

NEL Chapter 1 17

INVESTIGATE the Math

Two students created a game that they called “Which function am I?” In this game, players turn over cards that are placed face down and match the characteristics and properties with the correct functions. The winner is the player who has the most pairs at the end of the game.

The students have studied the following parent functions:

Properties of Graphs

of Functions

GOAL Compare and contrast the properties of various types of functions.

YOU WILL NEED

• graphing calculator

2 0

–4 –2 2 4

x

2 0

–4 –2 2 4

2 0

–4 –2 2 4

2 0

–4 –2 2 4

2 0

–4 –2 2 4

2 0

g ( x ) = x^2 h ( x ) = 1 x k ( x ) = | x |

2 0

–4 –2 2 4

p ( x ) = 2 x^ q ( x ) = sin x

f ( x ) = x

m ( x ) = x

Which criteria could the students use to differentiate between these different types of functions?

A. Graph each of these parent functions on a graphing calculator, and

sketch its graph. State the domain and range of each function, and determine its zeros and y -intercepts.

B. Determine the intervals of increase and the intervals of decrease for each of the parent functions.

?

interval of increase the interval(s) within a function’s domain, where the y -values of the function get larger, moving from left to right interval of decrease the interval(s) within a function’s domain, where the y -values of the function get smaller, moving from left to right

APPLY the Math

EXAMPLE 1 Connecting the graph of a function with its characteristics

2 0

–4 –2 2 4

x

2 0

–4 –2 2 4

f ( x ) = x f ( x ) = 2 x^ f ( x ) = | x |^ f ( x ) = sin^ ( x ) 2 0

–4 –2 2 4

2 0

Domain: { x PR}

Range: { y PR y $ 0}

As x 2`, y 0.

Infinite Number of Zeros

Match each parent function card with a characteristic of its graph. Each card may only be used for one parent function.

Solution

As x 2`, 2^ y^ 0. 0

• 4

–4 –2^24

f ( x ) = 2 x

This property describes the end behaviour: as x becomes negatively large, y approaches zero. The function must have a horizontal asymptote defined by The function must be y 5 2 x.

y 5 0.

Infinite Number of Zeros

2 0

–180˚^180 ˚

f ( x ) = sin x The sine function is periodic and continues infinitely, intersecting the x -axis an infinite number of times.

NEL Chapter 1 19

20 1.3^ Properties of Graphs of Functions NEL

State which of the parent functions in this lesson have the following characteristics: a) b) Solution a)

b)

q ( x ) 5 sin x

p ( x ) 5 2 x^ (Range 5 5 y P R 0 y $ 06 )

k ( x ) 5 Z x Z (Range 5 5 y P R 0 y $ 06 )

g ( x ) 5 x^2 (Range 5 5 y P R 0 y $ 06 )

f ( x ) 5 x (Range 5 5 y P R 6 )

Range 5 5 y P R Z 21 # y # 16

q ( x ) 5 sin x

p ( x ) 5 2 x

m ( x ) 5 " x (Domain 5 5 x P R 0 x $ 06 )

k ( x ) 5 Z x Z

h ( x ) 5

1 x (Domain^5 5 x P R^^0 x^^2 06 )

g ( x ) 5 x^2

f ( x ) 5 x

Domain 5 5 x P R 6

Range 5 5 y P R 0 21 # y # 16

Domain 5 5 x P R 6

EXAMPLE 2 Using reasoning to determine the equation of a parent function

There are five parent functions that match this characteristic and two that do not.

Of these five functions, only the sine function has the range 5 y P R 0 21 # y # 16.

There are five parent functions that match this characteristic and two that do not.

If you are given some characteristics of a function, you may be able to determine the equation of the function.

2 0

–4 –2^24

f ( x ) = | x | (^) Range: { y PR y $ 0}

The range indicates that all y -values of the function must be non-negative. This is true for both and However, has the x -axis as its horizontal asymptote, so y 2 0.Choose f ( x ) 5 0 x 0.

f ( x ) 5 0 x 0. f ( x ) 5 2 x

f ( x ) 5 2 x

5 y P R Z y $ 06

2 0

–4 –2 2 4

Domain: { x PR}

f ( x ) = x The remaining function, matches the property 5 x P R 6.

f ( x ) 5 x ,