1. Introduction to Functions
3.5/3.6
Introduction to Functions

2. Introduction to Functions
Define and identify relations and functions.
Relation
A relation is any set of ordered pairs.
A special kind of relation, called a function, is very important in mathemat-
ics and its applications.
Function
A function is a relation in which, for each value of the first component
of the ordered pairs, there is exactly one value of the second component.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

3. Introduction to Functions
EXAMPLE 1
Determining Whether Relations Are Functions
Tell whether each relation defines a function.
L = { (2, 3), (–5, 8), (4, 10) }
M = { (–3, 0), (–1, 4), (1, 7), (3, 7) }
N = { (6, 2), (–4, 4), (6, 5) }

4. Introduction to Functions
Mapping Relations F G 1 2 –1 6 –3 5 4 3 –2
F is a function.
G is not a function.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

5. Introduction to Functions
Tables and Graphs x y
x y –
2 –6 O
Table of the
function, F
Graph of the function, F
Copyright © 2010 Pearson Education, Inc. All rights reserved.

6. 4.5 Introduction to Functions
Domain and Range
In a relation, the set of all values of the independent variable (x) is the
domain. The set of all values of the dependent variable (y) is the range.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

7. 4.5 Introduction to Functions
EXAMPLE 2
Finding Domains and Ranges of Relations
Give the domain and range of each relation. Tell whether the relation defines
a function.
(a) { (3, –8), (5, 9), (5, 11), (8, 15) }
The domain, the set of x-values, is {3, 5, 8}; the range, the set of y-values,
is {–8, 9, 11, 15}.
This relation is not a function because the same x-value 5 is
paired with two different y-values, 9 and 11.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

8. 4.5 Introduction to Functions
EXAMPLE 2
Finding Domains and Ranges of Relations
Give the domain and range of each relation. Tell whether the relation defines
a function.
(b) 6 M 1 –9 N
The domain of this relation is {6, 1, –9}. The range is {M, N}.
This mapping defines a function – each x-value corresponds to exactly one
y-value.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

9. Introduction to Functions
EXAMPLE 2
Finding Domains and Ranges of Relations
Give the domain and range of each relation. Tell whether the relation defines
a function.
(c)
x y –
This is a table of ordered pairs, so the domain is the set of x-values,
{–2, 1, 2}, and the range is the set of y-values, {3}.
The table defines a
function because each different x-value corresponds to exactly one y-value
(even though it is the same y-value).
Copyright © 2010 Pearson Education, Inc. All rights reserved.

10. Introduction to Functions
EXAMPLE 3
Finding Domains and Ranges from Graphs
Give the domain and range of each relation.
(a) x y
The domain is the set of x-values,
{–3, 0, 2 , 4}. The range, the set of
y-values, is {–3, –1, 1, 2}.
(–3, 2)
(2, 1) O
(4, –1)
(0, –3)
Copyright © 2010 Pearson Education, Inc. All rights reserved.

11. Introduction to Functions
EXAMPLE 3
Finding Domains and Ranges from Graphs
Give the domain and range of each relation.
(b) x y
The x-values of the points on the
graph include all numbers between
–7 and 2, inclusive.
The y-values
include all numbers between –2 and
2, inclusive.
Using interval notation,
the domain is [–7, 2];
the range is [–2, 2].
Range O
Domain
Copyright © 2010 Pearson Education, Inc. All rights reserved.

12. Introduction to Functions
EXAMPLE 3
Finding Domains and Ranges from Graphs
Give the domain and range of each relation.
(c) x y
The arrowheads indicate that the
line extends indefinitely left and right,
as well as up and down. Therefore,
both the domain and range include
all real numbers, written (-∞, ∞). O
Copyright © 2010 Pearson Education, Inc. All rights reserved.

13. Introduction to Functions
EXAMPLE 3
Finding Domains and Ranges from Graphs
Give the domain and range of each relation. x y
(d)
The arrowheads indicate that the
graph extends indefinitely left and
right, as well as upward. The domain
is (-∞, ∞).
Because there is a least y-
value, –1, the range includes all
numbers greater than or equal to –1,
written [–1, ∞). O
Copyright © 2010 Pearson Education, Inc. All rights reserved.

14. 4.5 Introduction to Functions
Agreement on Domain
The domain of a relation is assumed to be all real numbers that produce
real numbers when substituted for the independent variable.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

15. 4.5 Introduction to Functions
Vertical Line Test
If every vertical line intersects the graph of a relation in no more than
one point, then the relation represents a function. y y
(a)
(b) x x
Not a function – the same
x-value corresponds to two
different y-values.
Function – each x-value
corresponds to only one
y-value.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

16. 4.5 Introduction to Functions
EXAMPLE 4
Using the Vertical Line Test
Use the vertical line test to determine whether each relation is a function.
(a) x y
This relation is a function.
(–3, 2)
(2, 1) O
(4, –1)
(0, –3)
Copyright © 2010 Pearson Education, Inc. All rights reserved.

17. Introduction to Functions
EXAMPLE 4
Using the Vertical Line Test
Use the vertical line test to determine whether each relation is a function.
(b) x y
This graph fails the vertical line test
since the same x-value corresponds
to two different y-values; therefore,
it is not the graph of a function. O
Copyright © 2010 Pearson Education, Inc. All rights reserved.

18. Introduction to Functions
EXAMPLE 4
Using the Vertical Line Test
Use the vertical line test to determine whether each relation is a function.
(c) x y
This relation is a function. O
Copyright © 2010 Pearson Education, Inc. All rights reserved.

19. Introduction to Functions
EXAMPLE 4
Using the Vertical Line Test
Use the vertical line test to determine whether each relation is a function.
(d) x y
This relation is a function. O
Copyright © 2010 Pearson Education, Inc. All rights reserved.

20. Introduction to Functions
EXAMPLE 5
Identifying Functions from Their Equations
Decide whether each relation defines a function and give the domain.
(a) y = x – 5
Copyright © 2010 Pearson Education, Inc. All rights reserved.

21. Introduction to Functions
EXAMPLE 5
Identifying Functions from Their Equations
Decide whether each relation defines a function and give the domain.
(b) y = 3x – 1

22. Introduction to Functions
EXAMPLE 5
Identifying Functions from Their Equations
Decide whether each relation defines a function and give the domain.
(e) y = 3
x + 4

23. Introduction to Functions
Variations of the Definition of Function
1. A function is a relation in which, for each value of the first component
of the ordered pairs, there is exactly one value of the second
component.
2. A function is a set of ordered pairs in which no first component is
repeated.
3. A function is a rule or correspondence that assigns exactly one range
value to each domain value.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

24. Introduction to Functions
Function Notation
When a function f is defined with a rule or an equation using x and y for the
independent and dependent variables, we say “y is a function of x” to
emphasize that y depends on x. We use the notation
y = f (x),
called function notation, to express this and read f (x), as “f of x”.
The letter f stands for function. For example, if y = 5x – 2, we can name
this function f and write
f (x) = 5x – 2.
Note that f (x) is just another name for the dependent variable y.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

25. Introduction to Functions
Function Notation
CAUTION
The symbol f (x) does not indicate “f times x,” but represents the y-value
for the indicated x-value. As shown below, f (3) is the y-value that
corresponds to the x-value 3.
y = f (x) = 5x – 2
y = f (3) = 5(3) – 2 = 13
Copyright © 2010 Pearson Education, Inc. All rights reserved.

26. Introduction to Functions
EXAMPLE 6
Using Function Notation
Let f (x) = x + 2x – 1. Find the following. 2
(a) f (4)
f (x) = x + 2x – 1 2
f (4) = • 4 – 1 2
Replace x with 4.
f (4) = – 1
f (4) = 23
Since f (4) = 23, the ordered pair (4, 23) belongs to f.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

27. Introduction to Functions
EXAMPLE 6
Using Function Notation
Let f (x) = x + 2x – 1. Find the following. 2
(b) f (w)
f (x) = x + 2x – 1 2
f (w) = w + 2w – 1 2
Replace x with w.
The replacement of one variable with another is important in later
courses.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

28. Introduction to Functions
EXAMPLE 7
Using Function Notation
Let g(x) = 5x + 6. Find and simplify g(n + 2).
g(x) = 5x + 6
g(n + 2) = 5(n + 2) + 6
Replace x with n + 2.
= 5n
= 5n + 16
Copyright © 2010 Pearson Education, Inc. All rights reserved.

29. Introduction to Functions
EXAMPLE 8
Using Function Notation
For each function, find f (7).
(a) f (x) = –x + 2
f (x) = –x + 2
f (7) = –7 + 2
Replace x with 7.
= –5
Copyright © 2010 Pearson Education, Inc. All rights reserved.

30. 4.5 Introduction to Functions
EXAMPLE 8
Using Function Notation
For each function, find f (7).
(b) f = {(–5, –9), (–1, –1), (3, 7), (7, 15), (11, 23)}
We want f (7), the y-value of the ordered pair
where x = 7.
As indicated by the ordered pair
(7, 15), when x = 7, y = 15, so f (7) = 15.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

31. 4.5 Introduction to Functions
EXAMPLE 8
Using Function Notation
For each function, find f (7).
(c) f
Domain
Range 4 11 7 7 17 17 10 23
The domain element 7 is paired with 17
in the range, so f (7) = 17.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

32. 4.5 Introduction to Functions
EXAMPLE 8
Using Function Notation
For each function, find f (7). x y
(d)
To evaluate f (7), find 7 on
the x-axis. 7
Then move up 5
until the graph of f is
reached.
tally to the y-axis gives 3
for the corresponding
y-value. Thus, f (7) = 3. 3
Moving horizon- 1 O 1 3 5 7
Copyright © 2010 Pearson Education, Inc. All rights reserved.

33. Introduction to Functions
Finding an Expression for f (x)
Step 1 Solve the equation for y.
Step 2 Replace y with f (x).
Copyright © 2010 Pearson Education, Inc. All rights reserved.

34. Introduction to Functions
EXAMPLE 9
Writing Equations Using Function Notation
Rewrite each equation using function notation. Then find f (–3) and f (n).
(a) y = x – 1 2
This equation is already solved for y. Since y = f (x),

35. Introduction to Functions
EXAMPLE 9
Writing Equations Using Function Notation
Rewrite each equation using function notation. Then find f (–3) and f (n).
(b) x – 5y = 3
First solve x – 5y = 3 for y. Then replace y with f (x).

36. Introduction to Functions
Linear Function
A function that can be defined by
f (x) = ax + b,
for real numbers a and b is a linear function. The value of a is the slope
of m of the graph of the function.
Copyright © 2010 Pearson Education, Inc. All rights reserved.