1. Introduction to Functions
3.6
Introduction to Functions 1
Understand the definition of a relation.
Understand the definition of a function.
Decide whether an equation defines a function.
Find domains and ranges.
Use function notation.
Apply the function concept in an application. 2 3 4 5 6

2. Understand the definition of a relation.
In an ordered pair (x, y), x and y are called the components of the ordered pair.
Any set of ordered pairs is called a relation.
The set of all first components of the ordered pairs of a relation is the domain of the relation, and the set of all second components of the ordered pairs is the range of the relation.

3. EXAMPLE 1
Identifying Domains and Ranges of Relations
Identify the domain and range of the relation.
Solution:
Domain:
Range:

4. Understand the definition of a function.
A very important type of relation called a function.
Function
A function is a set of ordered pairs in which each first component corresponds to exactly one second component.
By definition, the relation in the following order pairs is not a function, because the same first component, 3, corresponds to more then one second component.
If the ordered pairs from this example were interchanged, giving the relation
the result would be a function.
In that case, each domain element (first component) corresponds to exactly one range element (second component).

5. Mapping
Relations and functions can also be expressed as a correspondence or mapping from one set to another. In the example below the arrows from 1 to 2 indicates that the ordered pair (1, 2) belongs to F. Each first component is paired with exactly one second component.
x-axis values
y-axis values 1
– 2 3 2 4
– 1

6. Mapping
In the mapping for relations H, which is not a function, the first component – 2 is paired with two different second components, 1 and 0.
x-axis values
y-axis values
– 4
– 2 1

7. EXAMPLE 2
Determining Whether Relations Are Functions
Determine whether each relation is a function.
Solution: function
Solution: not a function

8. Decide whether an equation defines a function.
Given the graph of an equation, the definition of a function can be used to decide whether or not the graph represents a function. By the definition of a function, each x-value must lead to exactly one y-value.
Vertical Line Test
If a vertical line intersects a graph in more than one point, then the graph is not the graph of a function.
Any nonvertical line is the graph of a function. For this reason, any linear equation of the form y = mx + b defines a function. (Recall that a vertical line has an undefined slope.)

9. a. USING THE VERTICAL LINE TEST
Use the vertical line test to determine whether each relation graphed is a function. y a.
(1, 2)
This graph represents a function.
(– 1, 1) x
(0, – 1)
(4, – 3)

10. EXAMPLE 3
Determining Whether Relations Define Functions
Determine whether each relation is a function.
Solution: not a function
Solution: function

11. b. USING THE VERTICAL LINE TEST
Use the vertical line test to determine whether each relation graphed is a function. y b. 6
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. x
– 4 4
– 6

12. c. USING THE VERTICAL LINE TEST
Use the vertical line test to determine whether each relation graphed is a function. y c.
This graph represents a function. x

13. d. USING THE VERTICAL LINE TEST
Use the vertical line test to determine whether each relation graphed is a function. y d.
This graph represents a function. x

14. Find domains and ranges.
By the definitions of domain and range given for relations, the set of all numbers that can be used as replacements for x in a function is the domain of the function. The set of all possible values of y is the range of the function.

15. FINDING DOMAINS AND RANGES OF RELATIONS
Give the domain and range of the relation. Tell whether the relation defines a function. a.
The domain, the set of x-values, is {3, 4, 6}; the range, the set of y-values is {– 1, 2, 5, 8}. This relation is not a function because the same x-value, 4, is paired with two different y-values, 2 and 5.

16. FINDING DOMAINS AND RANGES OF RELATIONS
Give the domain and range of the relation. Tell whether the relation defines a function. b. 4 6 7
– 3
100
200
300
The domain is {4, 6, 7, – 3}; the range is {100, 200, 300}. This mapping defines a function. Each x-value corresponds to exactly one y-value.

17. FINDING DOMAINS AND RANGES OF RELATIONS
Give the domain and range of the relation. Tell whether the relation defines a function. c.
This relation is a set of ordered pairs, so the domain is the set of x-values {– 5, 0, 5} and the range is the set of y-values {2}. The table defines a function because each different x-value corresponds to exactly one y-value. x y
– 5 2 5

18. Give the domain and range of each relation.
FINDING DOMAINS AND RANGES FROM GRAPHS
Give the domain and range of each relation. y a.
The domain is the set of x-values which are
{– 1, 0, 1, 4}.
The range is the set of y-values which are
{– 3, – 1, 1, 2}.
(1, 2)
(– 1, 1) x
(0, – 1)
(4, – 3)

19. Give the domain and range of each relation.
FINDING DOMAINS AND RANGES FROM GRAPHS
Give the domain and range of each relation. y
The x-values of the points on the graph include all numbers between – 4 and 4, inclusive. The y-values include all numbers between – 6 and 6, inclusive. b. 6 x
– 4 4
The domain is [– 4, 4].
The range is [– 6, 6].
– 6

20. Give the domain and range of each relation.
FINDING DOMAINS AND RANGES FROM GRAPHS
Give the domain and range of each relation. y c.
The arrowheads indicate that the line extends indefinitely left and right, as well as up and down. Therefore, both the domain and the range include all real numbers, written
(– , ). x

21. Give the domain and range of each relation.
FINDING DOMAINS AND RANGES FROM GRAPHS
Give the domain and range of each relation. y d.
The arrowheads indicate that the line extends indefinitely left and right, as well as upward. The domain is (– , ). Because there is at least y-value, – 3, the range includes all numbers greater than, or equal to
– 3 or [– 3, ). x

22. EXAMPLE 4
Finding the Domain and Range of Functions
Find the domain and range of the function y = x2 + 4.
Solution:
Domain:
Range:

23. Use function notation.
The letters f, g, and h are commonly used to name functions. For example, the function y = 3x + 5 may be written
where f (x), which represents the value of f at x, is read “f of x.” The notation f (x) is another way of writing y in a function. For the function defined by f (x) = 3x + 5, if x = 7, then
Read this result, f (7) = 26, as “f of 7 equals 26.” The notation f (7) means the values of y when x is 7. The statement f (7) = 26 says that the value of y = 26 when x is 7. It also indicates that the point (7,26) lies on the graph of f.
The notation f (x) does not mean f times x; f (x) means the value of x for the function f. It represents the y –value that corresponds to x in the function f.

24. Use function notation. (cont’d)
In the notation f (x),
f is the name of the function,
x is the domain value,
and f (x) is the range value y for the domain value x.

25. f (x) = 6x − 2 EXAMPLE 5 Using Function Notation
Find f (−1), for the function.
f (x) = 6x − 2
Solution:

26. For each function, find (3).
USING FUNCTION NOTATION
For each function, find (3). b.
Solution For  = {( – 3, 5), (0, 3), (3, 1), (6, – 9)}, we want (3), the y-value of the ordered pair where x = 3. As indicated by the ordered pair (3, 1), when x = 3, y = 1,so(3) = 1.

27. Increasing, Decreasing, and Constant Functions
Suppose that a function  is defined over an interval I. If x1 and x2 are in I,
 increases on I if, whenever x1 < x2, (x1) < (x2)
 decreases on I if, whenever x1 < x2, (x1) > (x2)
 is constant on I if, for every x1 and x2, (x1) = (x2)

28. DETERMINING INTERVALS OVER WHICH A FUNCTION IS INCREASING, DECREASING, OR CONSTANT
Determine the intervals over which the function is increasing, decreasing, or constant. y 6 2 x
– 2 1 3

29. DETERMINING INTERVALS OVER WHICH A FUNCTION IS INCREASING, DECREASING, OR CONSTANT
Determine the intervals over which the function is increasing, decreasing, or constant. y
Solution 6
On the interval (– , 1), the y-values are decreasing; on the interval [1,3], the y-values are increasing; on the interval [3, ), the y-values are constant (and equal to 6). 2 x
– 2 1 3

30. EXAMPLE 6
Applying the Function Concept to Population
The median age at first marriage for women in the United States for selected years is given in the table.
Write a set of ordered pairs that defines a function f for these data.
Solution:
Give the domain and range of f.
Find f (2006).

31. Khan Academy Videos