1. Sullivan Algebra and Trigonometry: Section 3.1
Objectives
Determine Whether a Relation Represents a Function
Find the Value of a Function
Find the Domain of a Function
Identify the Graph of a Function
Obtain Information from or about the Graph of a Function

2. Let X and Y be two nonempty sets of real numbers
Let X and Y be two nonempty sets of real numbers. A function from X into Y is a rule or a correspondence that associates with each element of X a unique element of Y.
The set X is called the domain of the function.
For each element x in X, the corresponding element y in Y is called the image of x. The set of all images of the elements of the domain is called the range of the function.

3. f x y x y x X Y
RANGE
DOMAIN

4. {(1, 1), (2, 4), (3, 9), (-3, 9)} {(1, 1), (1, -1), (2, 4), (4, 9)}
Example: Which of the following relations are function?
{(1, 1), (2, 4), (3, 9), (-3, 9)}
A Function
{(1, 1), (1, -1), (2, 4), (4, 9)}
Not A Function

5. Functions are often denoted by letters such as f, F, g, G, and others
Functions are often denoted by letters such as f, F, g, G, and others. The symbol f(x), read “f of x” or “f at x”, is the number that results when x is given and the function f is applied.
Elements of the domain, x, can be though of as input and the result obtained when the function is applied can be though of as output.
Restrictions on this input/output machine:
1. It only accepts numbers from the domain of the function.
2. For each input, there is exactly one output (which may be repeated for different inputs).

6. Example: Given the function
Find:
f (x) is the number that results when the number x is applied to the rule for f.
Find:

7. The domain of a function f is the set of real numbers such that the rule of the function makes sense.
Domain can also be thought of as the set of all possible input for the function machine.
Example: Find the domain of the following function:
Domain: All real numbers

8. Example: Find the domain of the following function:

9. When a function is defined by an equation in x and y, the graph of the function is the graph of the equation, that is, the set of all points (x,y) in the xy-plane that satisfies the equation.
Vertical Line Test for Functions:
A set of points in the xy-plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.

10. Example: Does the following graph represent a function?y x
The graph does not represent a function, since it does not pass the vertical line test.

11. Example: Does the following graph represent a function?y x
The graph does represent a function, since it does passes the vertical line test.

12. Determine the domain, range, and intercepts of the following graph.y
(2, 3) 4
(10, 0)
(4, 0) x
(1, 0)
(0, -3) -4