Algebra 3

Lesson 3.1

Objective: SSBAT define and evaluate functions.

Standards: 2.2.11C, 2.8.11B,C,D

Function

A relation (set of ordered pairs) in which each number in the Domain is paired with exactly 1 number from the Range

i.e. A set of ordered pairs where no two pairs have the same x-value

The x-coordinates can NOT repeat

Function

{(3, 9), (-2, 4), (1, 1), (2, 4), (-4, 16)}

NOT a Function

{(0, 0), (4, 2), (1, -1), (4, -2)}

* 2 ordered pairs have the same x-value *

Examples: Are each of the following relations functions?

1. {(1,8), (2,7), (3,6), (4,5), (5,6)}

2. {(2,4), (1,10), (3,6), (1,5)}

Yes

No – the x-value of 1 as two output values

3.

x y

-5 8

-3 5

-1 8

0 5

2 8

4.

Yes No the x-value 4 has 3 different y-values

Determining if an equation represents a function

Solve the equation for y If you have to take an even root (, , etc) it is NOT a fuction Otherwise it is

Shortcut

If the equation has y to an Odd power it IS a function.y, y3, y5, etc.

If the equation has y to an Even power it’s NOT a functiony2 , y4 , y6 , etc.

1. x2 + y = 1

Yes – for each x there is only 1 y y = 1 – x2

2. y2 = x + 1

No – Each x has 2 possible y values

Determine if the following represent a function

Determining if a Graph represents a Function.

Use the Vertical Line Test

If a vertical line CAN pass through the graph, without touching it in more than one place at a time, it IS a function.

Examples: Determine if each represents a function.

1.

Function

2.

Not a Function

3.

Function

4.

Not A Function

5.

Not A Function

Determining if each is a function

Set of Ordered Pairs If the x-coordinates are all different it IS a function

Equation If the y has an odd exponent it IS a function

Graph If it passes the vertical line test it IS a function

Determine if each represents a function or not.

1. {(5, -2), (7, 0), (-3, 8), (6, 0)}

2. 5x – 4y3 = 9

3.

Function Notation

f(x)

Read as “f of x”

For functions, y and f(x) are the same thing

(just 2 different notations)

It does NOT mean f times x.

f(x) means we have a function, called f, that has the variable x.

Instead of saying:

y = 2x – 5 , solve for when x = 3

Function Notation allows us to write:

f(x) = 2x – 5 find f(3)

Evaluating Functions

Substitute the number that is in the parentheses in for the variable and solve

f(x) = 2x – 5 find f(3)

What it means: Let x = 3 and simplify the right side

(don’t do anything to the Left side)

f(3) = 2(3) – 5

= 6 – 5

= 1

So: f(3) = 1

Examples: Evaluate each.

1. f(x) = 3x – 4

Find f(5)

let x = 5 and solve

f(5) = 3(5) – 4

f(5) = 11

2. g(x) = x2 + 3x

Find g(-2)

g(-2) = (-2)2 + 3(-2)

g(-2) = -2

Just simplify the right side

3. f(x) = -2x – 11

Find f(-3)

f(-3) = -2(-3) – 11

= -5

4. f(x) = 8x + 5

Find f(x + 4)

f(x + 4) = 8(x + 4) + 5

= 8x + 32 + 5

= 8x + 37

5. g(x) = x2 + 7

Find g(x + 1)

g(x + 1) = (x + 1)2 + 7

(x + 1)(x + 1) + 7

x2 + 1x + 1x + 1 + 7

x2 + 2x + 1 + 7

g(x + 1) = x2 + 2x + 8

6. f(x) = 28

43

x

x

Find f(6)

f(6) =

=

=

7. f(x) = 8x – 10

Find f(11) + f(-3)

f(11) = 8(11) – 10

= 78

f(-3) = 8(-3) – 10

= -34

f(11) + f(-3) = 78 + -34 = 44

8. f(x) = 3x2

Find: 4f(5)

1st: Find f(5) f(5) = 3(5)2

= 75

2nd: Take 4 times 75 4 ∙ 75

= 300

Answer: 4f(5) = 300

9. g(x) = 2x2 + 8

Find

=

10. If f(x) = 13 – x and g(x) = 4x – 10 which is greater f(-7) or g(7)?

f(-7) = 20 g(7) = 18

20 > 18 Therefore…

f(-7) is Greater

On Your Own

Let: f(x) = x3 + 4

1. Find f(-6)

2. Find 3f(2)

Homework

Worksheet 3.1