1. 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. M: Mother Function Joe Samantha Anna Ian Chelsea George Laura Julie
Hilary
Barbara
Sue
Students
Mothers

5. M: Mother function
Domain of M = {Joe, Samantha, Anna, Ian, Chelsea, George}
Range of M = {Laura, Julie, Hilary, Barbara}
In function notation we write
M(Anna) = Julie
M(George) = Barbara
M(x)=Hilary indicates that x = Chelsea

6. The mother function M written as ordered pairs
M = {(Joe, Laura), (Samantha, Laura),
(Anna, Julie), (Ian, Julie), (Chelsea, Hillary),
(George, Barbara) }

7. For the function f below , evaluate f at the indicated values and find the domain and range of f
Domain of f
Range of f 10 11 12 13 14 15 16 1 2 3 4 5 6 7

8. Graphical Displays of Functions
Another way to depict a function whose ordered pairs are made up of numbers, is to display the ordered pairs via a graph on the coordinate plane, with the first elements of the ordered pairs graphed along the horizontal axis, and the second elements graphed along the vertical axis.

9. Functions defined by Rules
Let f be function, defined on the set of natural numbers, consisting of ordered pairs where the second element of the ordered pair is the square of the first element.
Some of the ordered pairs in f are
(1,1) (2,4), (3,9), (4,16),…….
f is best defined by the rule f(x) = x²

10. Function Notation f(x)
Functions defined on infinite sets are denoted by algebraic rules.
Examples of functions defined on all real numbers R.
f(x) = x²; g(x) = 2x - 1; h(x) = x³
The symbol f(x) represents the real number in the range of the function f corresponding to the domain value x.
The ordered pair (x,f(x)) belongs to the function f.

11. Evaluating functions

12. Find the domain of the following functions:B)
Domain is all real numbers but

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

14. Graph of a function
The graph of the function f(x) is the set of points (x,y) in the xy-plane that satisfy the relation y = f(x).
Example: The graph of the function
f(x) = 2x – 1 is the graph of the equation
y = 2x – 1, which is a line.

15. Domain and Range from the Graph of a function
Domain = {x / or }
Range = {y / or }

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

17. 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.

18. y x
Not a function.

19. y x
Function.

20. Is this a graph of a function?y 4
(2, 3)
(10, 0)
(4, 0)
(1, 0) x
(0, -3) -4

21. Even functions A function f is an even function if
for all values of x in the domain of f.
Example: is even because

22. Odd functions A function f is an odd function if
for all values of x in the domain of f.
Example: is odd because

23. Determine if the given functions are even or odd

24. Graphs of Even and Odd functions
The graph of an even function is symmetric with respect to the x-axis.
The graph of an odd function is symmetric with respect to the origin.

25. x 4 1 3 -1 2 -2 -4

27. Determine if the function is even or odd?

28. Determine if the function is even or odd?

29. Determine if the function is even or odd?

30. Increasing and Decreasing Functions
A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2 we have f(x1)<f(x2).
A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2 we have f(x1)>f(x2).
A function f is constant on an open interval I if, for all choices of x, the values f(x) are equal.

31. Local Maximum, Local Minimum
A function f has a local maximum at c if there is an open interval I containing c so that, for all x = c in I, f(x) < f(c). We call f(c) a local maximum of f.
A function f has a local minimum at c if there is an open interval I containing c so that, for all x = c in I, f(x) > f(c). We call f(c) a local minimum of f.

32. Determine where the following graph is increasing, decreasing and constant. Find local maxima and minima.

33. y 4
(2, 3)
(4, 0)
(1, 0) x
(10, -3)
(0, -3)
(7, -3) -4