Section P.3 – Functions and their Graphs

Functions

Algebraic Function

Can be written as finite sums, differences, multiples, quotients, and radicals involving xn.

Examples:Transcendental Function

A function that is not Algebraic.Examples:

A relation such that there is no more than one output for each input

A

B

C

W

Z

4

2

2 14

3 10xx

f x x x

g x

sin

ln

h x x

g x x

4 Examples of Functions

X Y

-3 1

-1 0

0 4

5 7

7 3

X Y

10 2

15 -5

18 -5

20 1

7 -5

These are all functions

because every x value has only one possible y

value

Every one of these functions is a

relation.

3 Examples of Non-Functions

X Y

0 4

1 10

2 11

1 -3

5 3

Every one of these non-functions is a

relation.

Not a function since x=1 can

be either y=10 or y=-3

Not a function since x=-4

can be either y=7 or y=1

Not a function since multiple x values have multiple y

values

The Vertical Line Test

If a vertical line intersects a curve more than once, it is not a function.

Use the vertical line test to decide which graphs are functions. Make sure to circle the functions.

The Vertical Line Test

If a vertical line intersects a curve more than once, it is not a function.

Use the vertical line test to decide which graphs are functions. Make sure to circle the functions.

Function Notation: f(x)Equations that are functions are typically written in a

different form than “y =.” Below is an example of function notation:

The equation above is read:

f of x equals the square root of x.

The first letter, in this case f, is the name of the function machine and the value inside the parentheses is the input. The expression to the right of the equal sign shows what the machine does to the input.

f x xDoes not stand for “f times x”

It does stand for “plug a value for x into a formula f”

Example

If g(x) = 2x + 3, find g(5).

5g 2 5 3

5 10 3g

5 13g

You want x=5 since g(x) was

changed to g(5)

When evaluating, do not write g(x)!

You wanted to find g(5). So the

complete final answer includes

g(5) not g(x)

Solving v Evaluating

23If 3, complete the following:f x x

a. Evaluate 3f b. Find if 5x f x Substitute and Evaluate

The input (or x) is 3.

Solve for x

The output is -5.

23 3 3 2

35 3x

No equal sign Equal sign

2 3

1

238 x

12 x

Number Sets

Counting numbers (maybe 0, 1, 2, 3, 4, and so on)

Natural Numbers:

Positive and negative counting numbers (-2, -1, 0, 1, 2, and so on)Integers:

a number that can be expressed as an integer fraction (-3/2, -1/3, 0, 1, 55/7, 22, and so on)

Rational Numbers:

a number that can NOT be expressed as an integer fraction (π, √2, and so on)

Irrational Numbers:NONE

Number Sets

The set of all rational and irrational numbers

Real Numbers:

Natural Numbers

Integers

Rational Numbers

Irrational

Numbers

Rea

l Num

ber

Ven

n D

iagr

am:

Set NotationNot Included

The interval does NOT include the endpoint(s)

Interval Notation Inequality Notation GraphParentheses

( )< Less than

> Greater thanOpen Dot

Included The interval does include the endpoint(s)

Interval Notation Inequality Notation GraphSquare Bracket

[ ]≤ Less than

≥ Greater thanClosed Dot

Graphically and algebraically represent the following:

All real numbers greater than 11

Graph:

Inequality:

Interval:

Example 1

10 11 12

11x

11,Infinity never ends. Thus we always

use parentheses to indicate there is no

endpoint.

Describe, graphically, and algebraically represent

the following:

Description:

Graph:

Interval:

Example 2

1 3 5

1 5x

1,5

All real numbers greater than or equal to 1 and less than 5

Describe and algebraically represent the

following:

Describe:

Inequality:

Symbolic:

Example 3

-2 1 4

2 or 4x x

, 2 4,

All real numbers less than -2 or greater than 4

The union or combination of the

two sets.

Domain and Range

DomainAll possible input values (usually x), which allows the function to work.

RangeAll possible output values (usually y), which result from using the function.

The domain and range help determine the window of a graph.

x y

f

Example 1

Domain: ,

Range: 25,

Domain: 8,2 2,9

Range: 7,8

1 9y x x

Describe the domain and range of both functions in interval notation:

Example 2

Find the domain and range of .

h t 4 3t

t -32 -20 -15 5 -4 0 1 2 3

h 10 8 7 -7 4 2 1 ER ER

0, DOMAIN: RANGE:

The range is clear from the graph and table.

The input to a square root function must be greater

than or equal to 0

4 3t 0

3t 4

t 43

Dividing by a negative switches

the sign

, 43

The domain is not obvious with the graph or table.

Piecewise Functions

For Piecewise Functions, different formulas are used in different regions of the domain.

Ex: An absolute value function can be written as a piecewise function:

if 0

if 0

x xx

x x

Example 1Write a piecewise function for each given graph.

f x

f x

g x

g x

7

if x 4

5

if x 4

12 x 1 if 0x

x 1 if 0x

Example 2

Rewrite as a piecewise function.

f x x 2 1

Find the x value of the vertex

Change the absolute values to parentheses. Plus make the one on

the left negative.

x -3 -2 -1 0 1 2 3 4

f(x) 6 5 4 3 2 1 2 3

f x

x 2 1

2 1x

if x 2

if x 2

Use a graph or table to help.

Basic Types of Transformations

( h, k ): The Key Point

y a f x h k

When negative, the original graph is flipped about

the x-axis

When negative, the original graph is flipped

about the y-axis

Horizontal shift of h units

Vertical shift of k units

Parent/Original Function: y f xA vertical stretch if

|a|>1and a vertical

compression if |a|<1

Transformation Example1xy

Shift the parent graph four units to the left and three

units down.

Description:

14 3xy

Use the graph of below to describe and sketch the graph of .

Composition of Functions

g xff

g

First Second

f g xOR

Substituting a function or it’s value into another function. There are two notations:

(inside parentheses always first)

Example 1

Let and . Find:

1gf 2 5g x x 2 3f x x

211 5g

4

1 5

44 2 3f

11

8 3

1 11f g

Substitute x=1 into g(x) first

Substitute the result into f(x)

last

1gf

4

Example 2

Let and . Find:

g f x 2 5g x x 2 3f x x

2 3f x x

22 3 52 3g xx

24 12 9 5x x 2 3 2 3 5x x

24 12 4g f x x x

Substitute x into f(x) first

Substitute the result into g(x) last

24 12 4x x

24 12 9 5x x

g f x

2 3x

Even v Odd Function

Symmetrical with respect to the y-axis.

Even Function

Tests…

Replacing x in the function by –x yields

an equivalent function.

Replacing x in the function by –x yields the opposite of the function.

Symmetrical with respect to the origin.

Odd Function

f x f x f x f x

ExampleIs the function odd, even, or neither? 4f x x

4f x x

4f x x

Check out the graph first. Test by Replacing x in the function by –x.

The equation is even.

An equivalent equation.

Delta xΔx stands for “the change in x.” It is a variable that

represents ONE unknown value. For example, if x1 = 5 and x2 = 7 then Δx = 7 – 5 = 2. Δx can be algebraically manipulated similarly to single letter variables. Simplify the following statements:

21. 5 1x 2. x x x

x

5 1 5 1x x

225 10 1x x 2

25 5 5 1x x x

x x xx

1

xx