1. Function
A function is a relation in which, for each distinct value of the first component of the ordered pair, there is exactly one value of the second component.

2. Decide whether the relation defines a function.
DECIDING WHETHER RELATIONS DEFINE FUNCTIONS
Example 1
Decide whether the relation defines a function.
Solution Relation F is a function, because for each different x-value there is exactly one y-value. We can show this correspondence as follows.
x-values of F
y-values of F

3. Decide whether the relation defines a function.
DECIDING WHETHER RELATIONS DEFINE FUNCTIONS
Example 1
Decide whether the relation defines a function.
Solution In relation H the last two ordered pairs have the same x-value paired with two different y-values, so H is a relation but not a function.
Different y-values
Not a function
Same x-values

4. 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

5. 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.

6. FINDING DOMAINS AND RANGES OF RELATIONS
Example 2
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.

7. FINDING DOMAINS AND RANGES OF RELATIONS
Example 2
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.

8. FINDING DOMAINS AND RANGES OF RELATIONS
Example 2
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

9. Vertical Line Test
If each vertical line intersects a graph in at most one point, then the graph is that of a function.

10. b. USING THE VERTICAL LINE TEST Example 4
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

11. d. USING THE VERTICAL LINE TEST Example 4
Use the vertical line test to determine whether each relation graphed is a function. y d.
This graph represents a function. x

12. b. Domain is IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Example 5
Decide whether each relation defines a function and give the domain and range. b.
Solution
Solve the inequality.
Add 1.
Divide by 2.
Domain is

13. b. IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Example 5
Decide whether each relation defines a function and give the domain and range. b.
Solution
Solve the inequality.
Add 1.
Divide by 2.
Because the radical is a non-negative number, as x takes values greater than or equal to ½ , the range is y ≥ 0 or

14. Function Notation
When a function  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.
called a function notation, to express this and read (x) as “ of x.” The letter is he name given to this function. For example, if y = 9x – 5, we can name the function  and write

15. Variations of the Definition of Function
A function is a relation in which, for each distinct value of the first component of the ordered pairs, there is exactly one value of the second component.
A function is a set of ordered pairs in which no first component is repeated.
A function is a rule or correspondence that assigns exactly one range value to each distinct domain value.

16. Finding an Expression for (x)
Consider an equation involving x
and y. Assume that y can be expressed as a function  of x.
To find an expression for (x):
Solve the equation for y.
Replace y with (x).

17. 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)