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Functions and Their Graphs

Jackie Nicholas

Janet Hunter

Jacqui Hargreaves

Mathematics Learning Centre

University of Sydney

NSW 2006

c ?1999 University of Sydney Mathematics Learning Centre, University of Sydneyi

Contents

1 Functions 1

1.1 What is a function? . .............................. 1

1.1.1 Definition of a function......................... 1

1.1.2 The Vertical Line Test......................... 2

1.1.3 Domain of a function .......................... 2

1.1.4 Range of a function . .......................... 2

1.2 Specifying or restricting the domain of a function . ............. 6

1.3 The absolute value function .......................... 7

1.4 Exercises..................................... 8

2 More about functions 11

2.1 Modifying functions by shifting........................ 11

2.1.1 Vertical shift .............................. 11

2.1.2 Horizontal shift............................. 11

2.2 Modifying functions by stretching . . ..................... 12

2.3 Modifying functions by reflections . . ..................... 13

2.3.1 Reflection in thex-axis......................... 13

2.3.2 Reflection in they-axis......................... 13

2.4 Other effects . .................................. 14

2.5 Combining effects . . .............................. 14

2.6 Graphing by addition of ordinates . . ..................... 16

2.7 Using graphs to solve equations........................ 17

2.8 Exercises..................................... 19

2.9 Even and odd functions . . .......................... 21

2.10 Increasing and decreasing functions . ..................... 23

2.11 Exercises..................................... 24

3 Piecewise functions and solving inequalities 27

3.1 Piecewise functions . .............................. 27

3.1.1 Restricting the domain......................... 27

3.2 Exercises..................................... 29

3.3 Inequalities . .................................. 32

3.4 Exercises..................................... 35

Mathematics Learning Centre, University of Sydneyii

4 Polynomials 36

4.1 Graphs of polynomials and their zeros.................... 36

4.1.1 Behaviour of polynomials when|x|is large . ............. 36

4.1.2 Polynomial equations and their roots ................. 37

4.1.3 Zeros of the quadratic polynomial . . ................. 37

4.1.4 Zeros of cubic polynomials . . ..................... 39

4.2 Polynomials of higher degree .......................... 41

4.3 Exercises..................................... 42

4.4 Factorising polynomials............................. 44

4.4.1 Dividing polynomials .......................... 44

4.4.2 The Remainder Theorem........................ 45

4.4.3 The Factor Theorem .......................... 46

4.5 Exercises..................................... 49

5 Solutions to exercises 50

XY 1 2 3 45
3 2 f XY 1 2 3 45
3 6 2 g Mathematics Learning Centre, University of Sydney1

1 Functions

In this Chapter we will cover various aspects of functions. We will look at the definition of a function, the domain and range of a function, what we mean by specifying the domain of a function and absolute value function.

1.1 What is a function?

1.1.1 Definition of a function

A functionffrom a set of elementsXto a set of elementsYis a rule that assigns to each elementxinXexactly one elementyinY. One way to demonstrate the meaning of this definition is by using arrow diagrams. f:X→Yis a function. Every element inXhas associated with it exactly one element ofY.g:X→Yis not a function. The ele- ment 1 in setXis assigned two elements,

5 and 6 in setY.

A function can also be described as a set of ordered pairs (x,y) such that for anyx-value in the set, there is only oney-value. This means that there cannot be any repeatedx-values with differenty-values. The examples above can be described by the following sets of ordered pairs.

F={(1,5),(3,3),(2,3),(4,2)}is a func-

tion.G={(1,5),(4,2),(2,3),(3,3),(1,6)}is not a function. The definition we have given is a general one. While in the examples we have used numbers as elements ofXandY, there is no reason why this must be so. However, in these notes we will only consider functions whereXandYare subsets of the real numbers. In this setting, we often describe a function using the rule,y=f(x), and create a graph of that function by plotting the ordered pairs (x,f(x)) on the Cartesian Plane. This graphical representation allows us to use a test to decide whether or not we have the graph of a function: The Vertical Line Test. 0 xy y 0 x Mathematics Learning Centre, University of Sydney2

1.1.2 The Vertical Line Test

The Vertical Line Test states that if it isnot possibleto draw a vertical line through a graph so that it cuts the graph in more than one point, then the graphisa function.

This is the graph of a function. All possi-

ble vertical lines will cut this graph only once.This is not the graph of a function. The vertical line we have drawn cuts the graph twice.

1.1.3 Domain of a function

For a functionf:X→Ythedomainoffis the setX.

This also corresponds to the set ofx-values when we describe a function as a set of ordered pairs (x,y). If only the ruley=f(x) is given, then the domain is taken to be the set of all realxfor which the function is defined. For example,y=⎷ xhas domain; all realx≥0. This is sometimes referred to as thenaturaldomain of the function.

1.1.4 Range of a function

For a functionf:X→Ytherangeoffis the set ofy-values such thaty=f(x) for somexinX. This corresponds to the set ofy-values when we describe a function as a set of ordered pairs (x,y). The functiony=⎷ xhas range; all realy≥0.

Example

a.State the domain and range ofy=⎷ x+4. b.Sketch, showing significant features, the graph ofy=⎷ x+4. -4 -2 -1 xy 101
-33 1 -2 xy 02468
-1 -2 -3 Mathematics Learning Centre, University of Sydney3

Solution

a.The domain ofy=⎷ x+ 4 is all realx≥-4. We know that square root functions are only defined for positive numbers so we require thatx+4≥0, iex≥-4. We also know that the square root functions are always positive so the range ofy=⎷ x+4is all realy≥0. b.

The graph ofy=⎷

x+4.

Example

a.State the equation of the parabola sketched below, which has vertex (3,-3). b.Find the domain and range of this function.

Solution

a.The equation of the parabola isy= x 2 -6x 3 b.The domain of this parabola is all realx. The range is all realy≥-3.

Example

Sketchx

2 +y 2 = 16 and explain why it is not the graph of a function.

Solution

x 2 +y 2 = 16 is not a function as it fails the vertical line test. For example, whenx=0 y=±4. 24
-4y 0 -224-2 x -4

0213-112

x y Mathematics Learning Centre, University of Sydney4

The graph ofx

2 +y 2 = 16.

Example

Sketch the graph off(x)=3x-x

2 and find a.the domain and range b.f(q) c.f(x 2 d. f(2+h)-f(2) h ,h?=0.

Solution

The graph off(x)=3x-x

2 b.f(q)=3q-q 2 -2 x 246
y 024
Mathematics Learning Centre, University of Sydney5 c.f(x 2 )=3(x 2 )-(x 2 2 =3x 2 -x 4 d. f(2 +h)-f(2) h=(3(2 +h)-(2 +h) 2 )-(3(2)-(2) 2 h

6+3h-(h

2 +4h+4)-2 h -h 2 -h h =-h-1

Example

Sketch the graph of the functionf(x)=(x-1)

2 + 1 and show thatf(p)=f(2-p). Illustrate this result on your graph by choosing one value ofp.

Solution

The graph off(x)=(x-1)

2 +1. f(2-p) = ((2-p)-1) 2 +1 =(1-p) 2 +1 =(p-1) 2 +1 =f(p) -2 x 246
y 024
-1 x 24
y 021
Mathematics Learning Centre, University of Sydney6 The sketch illustrates the relationshipf(p)=f(2-p) forp=-1. Ifp=-1 then

2-p=2-(-1) = 3, andf(-1) =f(3).

1.2 Specifying or restricting the domain of a function

We sometimes give the ruley=f(x) along with the domain of definition. This domain may not necessarily be the natural domain. For example, if we have the function y=x 2 illustrate this by sketching the graph.

The graph ofy=x

2 04-2 |-2|=2 |4|=4 ax| | = | |a - x x - a y = -x x < 0y 1 12 0 -1x

2-2y = x x

0 Mathematics Learning Centre, University of Sydney7

1.3 The absolute value function

Before we define the absolute value function we will review the definition of the absolute value of a number. TheAbsolute value of a numberxis written|x|and is defined as |x|=xifx≥0or|x|=-xifx<0. That is,|4|= 4 since 4 is positive, but|-2|= 2 since-2 is negative. We can also think of|x|geometrically as the distance ofxfrom 0 on the number line. More generally,|x-a|can be thought of as the distance ofxfromaon the numberline.

Note that|a-x|=|x-a|.

The absolute valuefunctionis written asy=|x|.

We define this function as

y= +xifx≥0 -xifx<0 From this definition we can graph the function by taking each part separately. The graph ofy=|x|is given below.

The graph ofy=|x|.

y 1 12 0 x

2y = x - 2 x

2 y = -x + 2 x < 2

34
Mathematics Learning Centre, University of Sydney8

Example

Sketch the graph ofy=|x-2|.

Solution

Fory=|x-2|we have

y= +(x-2) whenx-2≥0orx≥2 -(x-2) whenx-2<0orx<2

That is,

y= x-2 forx≥2 -x+ 2 forx<2

Hence we can draw the graph in two parts.

The graph ofy=|x-2|.

We could have sketched this graph by first of all sketching the graph ofy=x-2 and then reflecting the negative part in thex-axis. We will use this fact to sketch graphs of this type in Chapter 2.

1.4 Exercises

1. a.State the domain and range off(x)=⎷9-x

2 b.Sketch the graph ofy=⎷ 9-x 2

2.Givenψ(x)=x

2 + 5, find, in simplest form,ψ(x+h)-ψ(x) hh?=0.

3.Sketch the following functions stating the domain and range of each:

Mathematics Learning Centre, University of Sydney9 a.y=⎷ x-1 b.y=|2x| c.y= 1 x-4 d.y=|2x|-1.

4. a.Find the perpendicular distance from (0,0) to the linex+y+k=0

b.If the linex+y+k= 0 cuts the circlex 2 +y 2 = 4 in two distinct points, find the restrictions onk.

5.Sketch the following, showing their important features.

a.y= 1 2 x b.y 2 =x 2

6.Explain the meanings of function, domain and range. Discuss whether or noty

2 =x 3 is a function.

7.Sketch the following relations, showing all intercepts and features. State which ones

are functions giving their domain and range. a.y=-⎷ 4-x 2 b.|x|-|y|=0 c.y=x 3 d.y= x |x| ,x?=0 e.|y|=x.

8.IfA(x)=x

2 +2+ 1 x 2 ,x?= 0, prove thatA(p)=A( 1 p ) for allp?=0.

9.Write down the values ofxwhich are not in the domain of the following functions:

a.f(x)=⎷ x 2 -4x b.g(x)= x x 2 -1

10.Ifφ(x) = log

x x-1 , find in simplest form: a.φ(3) +φ(4) +φ(5) b.φ(3) +φ(4) +φ(5) +···+φ(n)

11. a.Ify=x

2 +2xandx=(z-2) 2 , findywhenz=3. b.GivenL(x)=2x+ 1 andM(x)=x 2 -x, find i.L(M(x)) ii.M(L(x)) -1 1xy 2 0 -2 (-2,1) 0 x -2 y 2 15 (2,1) Mathematics Learning Centre, University of Sydney10

12.Using the sketches, find the value(s) of the constants in the given equations:

y=ax 2 -by= a bx 2 +1

13. a.Define|a|, the absolute value ofa, whereais real.

b.Sketch the relation|x|+|y|=1.

14.Given thatS(n)=

n 2n+1 , find an expression forS(n-1).

Hence show thatS(n)-S(n-1) =

1 (2n-1)(2n+1) 2 y

1-1-2 03

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