U2-S1-L2 Evaluating Functions

Essential Question:How and why do you evaluate functions?

Definitions

Relation A relationship between sets of information. Typically between inputs and outputs.

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

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

3 Examples of Non-Functions

X Y

0 4

1 10

2 11

1 -3

5 3

Function NotationThe 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.

25f

Which do you prefer to write?

OREvaluate f when x

= 25?

25

5

Example

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

5f 2 5 3

5 10 3f

5 13f

You want x=5 since f(x) was changed to

f(5)

When evaluating, do not write f(x)!

You wanted to find f(5). So the complete final

answer includes f(5) not f(x)

Function Notation – a function that uses f(x) for the output.

The notation g(x) and h(x) also indicate functions of x

f(x) is pronounced “f of x” or “f is a function of x”

Input values are called the independent variables

Output values are called the dependent variables

Example: y = 3x + 2 becomes f(x) = 3x + 2

EXAMPLE1a Evaluating a function rule

f(x) = –3x – 10

f(6) = –3(6) – 10

f(6) = –18 – 10

f(6) = –28

Evaluate f(x) = –3x – 10 for x = 6

EXAMPLE1b Evaluating a function rule

f(a) = –3a + 5

f(-3) = –3(–3) + 5

f(-3) = 9 + 5

f(6) = 14

f(1) = –3 + 5

f(1) = 2

f(a) = –3a + 5

f(4) = –3(4) + 5

f(6) = –12 + 5

f(6) = –7

Evaluate the function rule f(a) = –3a + 5 to find the range of the function for the domain {-3, 1, 4}

f(1) = –3(1) + 5

f(a) = –3a + 5

Check for Understanding

How would you explain to another student the process for evaluating expressions?

Write your steps in your notes.

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

Evaluate Functions from a Graph

X f(x)

-3 1

-1 0

0 4

5 7

7 3 You can evaluate a function from a table or graph as well by locating the correct input or output value.

Find f(0) for each function.

Find f(-1) for each function.

Find f(x) = 3 for each function.

Input

Output

FunctionRule

Function Rule – an equation that describes a function

Domain – is the set of input values

Range – is the set of output values

x = 6x = 4y = 10

x = 2

y = 3x + 4

output input

Input Output

x y

2 10

6 22

y = 16

4 16

y = 223x + 4

Input Output

x y

6 22 4 16 2 10

EXAMPLE3 Application

Suppose your group recorded a CD, now you want to copy and sell it. One company charges $250 plus $3 per CD. The total cost P(c) depends on the number of CDs burned. P(c) = 250 + 3c

C P(c) = 250 + 3c (c, P(c))

100 250 + 3(100) = 550 (100, 550)

200 250 + 3(200) = 850 (200, 850)

300 250 + 3(300) = 1150 (300, 1150)

500 250 + 3(500) = 1750 (500, 1750)

200 400 600

500

1000

1500

P(c)

Number of CDs

EXAMPLE4 Graphing Functions

Graph the function y = │x│ + 1. Find the domain and range.

x y DomainAll real numbers

Range{y: y > 1}

Evaluating Functions

Evaluating functions helps us find values for solving and graphing.

These are solutions to the equation, and points on the graph.

Summary

Answer the essential question in detailed, complete sentences. How and why do you evaluate functions?How and why do you evaluate functions?

Write 3-5 study questions in the left column to support the notes.