1. Functions 12-4 Warm Up Problem of the Day Lesson Presentation
Pre-Algebra

2. Pre-Algebra
12-4
Functions
Warm Up
1. Give the next three terms in each sequence using the simplest rule you can find.
1, 3, 1, 5, 1, 7, 1, 9, 1, 11, . . .
Find the first five terms of the sequence defined by the given rule.
2. an =
3. an = n(n – 1)
1, 13, 1, 15 2n
n + 1
1, , , , 4 3 2 8 5
0, 2, 6, 12, 20

3. What is the 1,111,111th term of the geometric sequence defined by
Problem of the Day
What is the 1,111,111th term of the geometric sequence defined by
a1 = 1, r = –1?
1; every odd term is 1, and every even term is –1.

4. Learn to represent functions with tables, graphs, or equations.

5. Vocabulary
function
input
output
domain
range
function notation

6. A function is a rule that relates two quantities so that each input value corresponds to exactly one output value.
The domain is the set of all possible input values, and the range is the set of all possible output values.

7. Functions can be represented in many ways, including tables, graphs, and equations. If the domain of a function has infinitely many values, it is impossible to represent them all in a table, but a table can be used to show some of the values and to help in creating a graph.

8. Additional Example 1: Finding Different Representations of a Function
Make a table and a graph of y = 3 – x2.
Make a table of inputs and outputs. Use the table to make a graph. x
3 – x2 y –2 –1 1 2
3 – (–2)2 –1
3 – (–1)2 2
3 – (0)2 3
3 – (1)2 2
3 – (2)2 –1

9. Make a table and a graph of y = x + 1.
Try This: Example 1
Make a table and a graph of y = x + 1.
Make a table of inputs and outputs. Use the table to make a graph. x y 2 3 –3 x
x + 1 y –1 1 2
–1 + 1
0 + 1 1
1 + 1 2
2 + 1 3

10. To determine if a relationship is a function, verify that each input has exactly one output.

11. Additional Example 2A: Identifying Functions
Determine if the relationship represents a function. A. x y 2 3 4 5 6
The input x = 2 has two outputs, y = 3 and y = 6. The input x = 3 also has more than one output. The relationship is not a function.

12. Additional Example 2B: Identifying Functions
Determine if the relationship represents a function. B.
The input x = 0 has two outputs, y = 2 and y = –2. Other x-values also have more than one y-value. The relationship is not a function.

13. Additional Example 2C: Identifying Functions
Determine if the relationship represents a function.
C. y = x3
Make an input-output table and use it to graph y = x3. x y –2
(–2)3 = –8 –1
(–1)3 = –1
(0)3 = 0 1
(1)3 = 1 2
(2)3 = 8
Each input x has only one output y. The relationship is a function.

14. Try This: Example 2A
Determine if the relationship represents a function. A. x y 1 2 3
Each input x has only one output y. The relationship is a function.

15. Determine if the relationship represents a function. B.
Try This: Example 2B
Determine if the relationship represents a function. B. x y
Since the relationship is linear there can only be one output y for each input x. The relationship is a function.

16. Try This: Example 2C
Determine if the relationship represents a function.
C. y = x – 1 x
x – 1 y
0 – 1
– 1 1
1 – 1 2
2 – 1 3
3 – 1
Each input x has only one output y. The relationship is a function.

17. You can describe a function using function notation
You can describe a function using function notation. In function notation, the output value of the function f that corresponds to the input value x is written as f(x). The expression f(x) means “the rule of f applied to the value of x,” not “f multiplied by x.”
f(x) is read “f of x.”
f(1) is read “f of 1.”
Reading Math

18. Additional Example 3A: Evaluating Functions
For the function, find f(0), f(-2), and f(1).
A. y = 3x + 2
f(x) = 3x + 2
Write in function notation.
f(0) = 3(0) + 2 = 2
f(–2) = 3(–2) + 2 = –4
f(1) = 3(1) + 2 = 5

19. Additional Example 3B: Evaluating Functions
For the function, find f(0), f(-2), and f(1). B.
Read the graph to find y for each x.
f(x) = y
f(0) = –1
f(–2) = 1
f(1) = –2

20. Additional Example 3C: Evaluating Functions
For the function, find f(0), f(-2), and f(1). C. x y –2 6 –1 5 4 1 3 2
Read the table to find y for each x.
f(x) = y
f(0) = 4
f(–2) = 6
f(1) = 3

21. Try This: Example 3A
For the function, find f(0), f(-1), and f(1).
A. y = 2x + 2
f(x) = 2x + 2
Write in function notation.
f(0) = 2(0) + 2 = 2
f(–1) = 2(–1) + 2 = 0
f(1) = 2(1) + 2 = 4

22. Try This: Example 3B
For each relationship, find f(–3), f(-1), and f(2). B.
Read the graph to find y for each x.
f(x) = y
f(–3) = 2
f(–2) = 1
f(2) = –3

23. Try This: Example 3C
For the function, find f(0), f(-1), and f(1). C.
Read the table to find y for each x. x y –2 4 –1 1 2 8
f(x) = y
f(0) = 0
f(–1) = 4
f(1) = 2

24. 1. Graph the function y = x2 – 3.
Lesson Quiz: Part 1
1. Graph the function y = x2 – 3. x y 2 -2 4 -4

25. Lesson Quiz: Part 2
Determine if each relationship represents a function. 2.
3. y = 3x + 5
4. For the function y = 3x2 – 2, find f(0), f(2), and f(–4). no
yes
–2, 10, 46