Warm Up

1. Find a triple if r = 10 and s = 2.2. If a = 6, b = 8 and the triangle is a

right triangle, what is the value of c? 3. If c = 50 and b = 48, what is the value of a if it is a right

triangle? 1

32

Lesson 4 Functions and Function Notation

Objective: To learn about functions, domains and ranges. To use function notation to represent functions

Functions

y = 3x + 1

For this equation, every x we choose, will give us a new y.

x is the input, and y is the output. For every x we choose there is one

and only one y possible – therefore this is a FUNCTION

Definition of a Function

A function is a correspondence between two sets X and Y that assigns to each element x of set X exactly one element y of set Y

Domain and Range

For each element x in X, the corresponding element y in Y is called the image of the function at x. The set X is called the domain of the function, and the set of all function values, Y is called the range of the function.

Ex 1: Determine whether each relation is a function.

a. {(4,5), (6,7), (8,8)}

b. {(5,6), (4,7), (6,6), (6,7)} We begin by making a figure

for each relation that shows set , the

domain, and set , t

So

he

luti

ra

o

e.

n

ng

X

Y

Solution for part (a)

468

578

X Y

Domain Range

The figure shows that every

element in the domain

corresponds to exactly one

element in the range.No two ordered pairs in the given relation have

the same first component different second

components. Thus, the relation is a function

Solution for part (b)

456

6

7

X Y

Domain Range

The figure shows that 6

corresponds to both 6 and 7.

If any element in the domain corresponds to

more than one element in the range, the

relation is not a function, Thus, the relation

is not a function.

Vertical Line Test (Pencil Test)

Given a graph – any vertical line drawn on the graph should only go through one point.

FunctionNot a Function

Function Notation

( )

The variable is called the

, because it can be assigned any

of the permissible numbers from the domain.

The variable is called the

independent

variable

dependent

, because

var its iable value depe

y f x

x

y

nds on .x

Function Notation

The special notation f(x), read “f of x”, represents the value of the function at the number x.

The notation does not mean “ f times x.”

Finding a Function’s Domain

The domain (the x’s) of a function can be almost any real number. There are two cases when there are exceptions: Division by zero – if x is in the bottom

of the fraction (denominator), then x cannot make that denominator 0.

f(x) = ; x = 3 is not defined. 1

3x

Finding a Function’s Domain

Even roots of negative numbers y = is only a real number if x ≥

11x

Finding a Function’s Domain

Exclude from a function's domain real

numbers that cause division by zero

and real numbers that result in an even

root of a negative number.

Ex 5: Find the domain of each function

2a. ( ) 8 5 2

2b. ( )

5

c. ( ) 2

f x x x

g xx

h x x

Solution part a

2The function ( ) 8 5 2 contains

neither division nor an even root.

The domain of is the set of all real numbers.

f x x x

f

Solution part b

2The function ( ) contains division.

5Because division by zero is undefined, we

must exclude from the domain values of

that cause 5 to be 0. Thus cannot equal

to 5. The domain of function is

g xx

x

x x

g

{ | 5}.x x

Solution part c

The function ( ) 2 contains an even

root. Because only non-negative numbers

have real square roots, the quantity under the

radical sign, 2 must be greater than or

equal to 0. Thus, 2 0 or 2

Th

h x x

x

x x

erefore the domain of is { | 2}

or the interval [ 2, ).

h x x

Practice Exercises

2

Find the domain of each function:

121. ( )

361

2. ( )2

xh x

x

f xx

Answers

1. { | 6, 6}x x x

2. { | 2} or ( 2, )x x

Graphing Functions

Using a graphing calculator Put the equation or function into the

form y = …. Button at top left y = Then type in the rest of the equation

and press [GRAPH] at top right.

Evaluating Functions

Given  f(x) = x2 + 2x – 1, find  f(2). This means to plug in 2 wherever

there is an x in the function. f(2) = (2)2 +2(2) – 1

       = 4 + 4 – 1        = 7

Evaluating Functions

Given  f(x) = x2 + 2x – 1, find  f(–3). f(–3) = (–3)2 +2(–3) – 1

         = 9 – 6 – 1          = 2

Practice

f(x) = x2 + 3x- 4, find f(-2) & f(0)

f(x) = x2 + 1, find f(-3) & f(x-1)

Practice

Find the numbers for x whose image is 2. f(x) = x2