Functions and Their Graphs

Functions and Their Graphs JackieNicholas JanetHunter JacquiHargreaves Mathematics Learning Centre University of Sydney

NSW 2006

cff1999 University of Sydney 1 / 4

Mathematics Learning Centre, University of Sydneyi Contents

• 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

• 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

• 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 2 / 4

Mathematics Learning Centre, University of Sydneyii

• 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

• Solutions to exercises 50 3 / 4

XY 1 2 3 4 5 3 2 f XY 12 3 4 5 3 6 2 g Mathematics Learning Centre, University of Sydney1

• 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,

• 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.

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