1. Introduction to Functions

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. Determine which of the following relations represent functions.
Not a function.
Function.
Function.

5. Not a function.
(2,1) and (2,-9)both work.

6. Theorem Vertical Line Test
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.

7. y x
Not a function.

8. y x
Function.

9. Interval Notation
A parenthesis ( ) shows an open (not included) endpoint
A bracket [ ] shows a closed [included] endpoint
Examples:
Set A with endpoints 1 and 3, neither endpoint included (1,3)
Set B with endpoints 6 and 10, not including [6,10)
Set C with endpoints 20 and 25, including both endpoints [20,25]
Set D with endpoints 28 and infinity, not including (28, )

10. Even functions are symmetric about the y-axis
What do you notice about the graphs of even functions?
Even functions are symmetric about the y-axis

11. Odd functions are symmetric about the origin
What do you notice about the graphs of odd functions?
Odd functions are symmetric about the origin

13. A function is odd if f( -x) = - f(x) for every number x in the domain.
A function is even if f( -x) = f(x) for every number x in the domain.
So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd.
EVEN
ODD

14. If a function is not even or odd we just say neither (meaning neither even nor odd)
Determine if the following functions are even, odd or neither.
Not the original and all terms didn’t change signs, so NEITHER.
Got f(x) back so EVEN.

15. Ex. 1
Even, Odd or Neither?
Graphically
Algebraically
EVEN

16. Ex. 2
Even, Odd or Neither?
Graphically
Algebraically
ODD

17. Ex. 3
Even, Odd or Neither?
Graphically
Algebraically
EVEN

18. Ex. 4
Even, Odd or Neither?
Graphically
Algebraically
Neither

19. Even, Odd or Neither?
EVEN
ODD

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

21. Find the domain of the following functions:B)

22. Square root is real only for nonnegative numbers.C)
Square root is real only for nonnegative numbers.