1. Functions
Definition: A function from a set D to a set R is a rule that assigns to every element in D a unique element in R.
Familiar Definition: For every input (x) there is only one output (y).
x: The independent variable
y: The dependent variable

2. Functions
To be a function a graph must pass the vertical line test which states a graph (set of points (x,y)) in the xy-plane defines y as a function of x iff no vertical line intersects the graph in more than one point.

3. Functions Which of the following are functions? a. b. c. d. Function
Not
Not
Function

4. Domain and Range
Definitions: Given a function , the DOMAIN of the function is the set of all permissible inputs and the RANGE is the set of all resulting outputs.
Domains can be found algebraically; ranges are found algebraically or graphically. Domains and Ranges are sets. Therefore, you must use the proper notation.

5. Domain and Range Proper Notation: { } – set (of intervals)
[ , ] – interval includes the endpoints
( , ) – interval does not include the endpoints
- the union (of intervals)

6. Finding the Domain
For polynomials the domain is the set of all real numbers R.
Square root functions can not be negative, so set the expression under the radical ≥0 and solve. This will be your domain.
Rational functions can not have zero in the denominator. Set the denominator ≠ 0. This will be the exclusion from the domain.

7. Find the Domain Determine the domain for the following functions: 1.2.
Domain = _____________ 3.
R or (–∞, ∞)
(–∞, 2/3 ]
(–∞ , –2) U (–2, 2) U (2, ∞)

8. Find the Range With the aid of a graphing calculator find the range of
Domain:
No zero in denom.
Range:
It also appears zero is not in the range. (Function itself can never equal zero since the numerator is a constant). Remember, you are interested in the vertical span of the graph.
(-∞,0) (0,∞)
(-∞,0) (0,∞)

9. Practice – Find the Domain