1. 3
Chapter
Chapter 2
Graphing

2. Introduction to Functions
Section
3.6
Introduction to Functions

3. Identifying Relations, Domains, and Ranges
Objective 1
Identifying Relations, Domains, and Ranges

4. Vocabulary
An equation in 2 variables defines a relation between the two variables.
A set of ordered pairs is also called a relation.
The domain is the set of x-coordinates of the ordered pairs.
The range is the set of y-coordinates of the ordered pairs.

5. Example
Find the domain and range of the relation {(4,9), (–4,9), (2,3), (10, –5)}.
Domain is the set of all x-values; {4, –4, 2, 10}
Range is the set of all y-values; {9, 3, –5}

6. Identifying Functions
Objective 2
Identifying Functions

7. Functions Some relations are also functions.
A function is a set of order pairs in which each
x-coordinate has exactly one y-coordinate.

8. Example
Is the relation {(4,9), (–4,9), (2,3), (10, –5)}, also a function?
Since each element of the domain is paired with only one element of the range, it is a function.
Note: It’s okay for a y-value to be assigned to more than one x-value, but an x-value cannot be assigned to more than one y-value (has to be assigned to ONLY one y-value).

9. Using the Vertical Line Test
Objective 3
Using the Vertical Line Test

10. Vertical Line Test
Graphs can be used to determine if a relation is a function.
Vertical Line Test
If a vertical line can be drawn so that it intersects a graph more than once, the graph is not the graph of a function.
(If no such vertical line can be drawn, the graph is that of a function.)

11. Example x y
Use the vertical line test to determine whether the graph to the right is the graph of a function.
Since no vertical line will intersect this graph more than once, it is the graph of a function.

12. Example x y
Use the vertical line test to determine whether the graph to the right is the graph of a function.
Since no vertical line will intersect this graph more than once, it is the graph of a function.

13. Example
Use the vertical line test to determine whether the graph to the right is the graph of a function. x y
Since vertical lines can be drawn that intersect the graph in two points, it is NOT the graph of a function.

14. Vertical Line Test
Since the graph of a linear equation is a line, all linear equations are functions, except those whose graph is a vertical line.
Thus, all linear equations are functions except those of the form x = c, which are vertical lines.

15. Example
Which of the following equations are functions? a. y = 2x b. y = –3x – 1 c. y = 8 d. x = 2
Function
Function
Function
NOT a Function

16. Using Function Notation
Objective 4
Using Function Notation

17. Using Function Notation
The variable y is a function of the variable x. For each value of x, there is only one value of y. Thus, we say the variable x is the independent variable because any value in the domain can be assigned to x. The variable y is the dependent variable because its value depends on x.
We often use letters such as f, g, and h to name functions. For example, the symbol f(x) means function of x and is read “f of x.” This notation is called function notation.
We can use function notation to write the equation
y = –3x + 2 as f(x) = –3x + 2.

18. Helpful Hint
Note that f(x) is a special symbol in mathematics used to denote a function. The symbol f(x) is read “f of x.” It does not mean f x (f times x).

19. Function Notation
When we want to evaluate a function at a particular value of x, we substitute the x-value into the notation.
For example, f(2) means to evaluate the function f when x = 2. So we replace x with 2 in the equation.
For our previous example when f(x) = –3x + 2, f(2) = –3(2) + 2 = –6 + 2 = –4.
When x = 2, then f(x) = –4, giving us the order pair
(2, –4).

20. Example
Given g(x) = x2 – 2x, find g(–3). Then write down the corresponding ordered pair.
g(–3) = (–3)2 – 2(–3) = 9 – (–6) = 15.
The ordered pair is (–3, 15).

21. Example Find the domain and the range of the function.y
Domain: [–3 ≤ x ≤ 4]
Domain
Find the domain and the range of the function.
Range: [–4 ≤ x ≤ 2]
Range

22. Example Find the domain and the range of the function graphed.y
Range: y ≥ –2
Range
Domain: all real numbers
Domain