1. Functions

2. Objectives Define a relation Define a function
Define the domain and range of a function
Evaluate a function
Graph a function

3. Set of Ordered Pairs
The table shows the number of medals won by United States athletes during five Winter Olympics.
The following is a set of ordered pairs: {(x, y) | x = year, y = number of medals} {(1992, 11), (1994, 13), (1998, 13), (2002, 34), (2006, 25), (2010, 37)}

4. Domain and Range of a Relation
Relation: set of ordered pairs {(1992, 11), (1994, 13), (1998, 13), (2002, 34), (2006, 25), (2010, 37)}
Domain: set of all first components of a relation
{1992, 1994, 1998, 2002, 2006, 2010}
Range: set of all second components of a relation
{11, 13, 34, 25, 37}

5. Example Given the relation {(–2, –5), (4, 7), (8, 9)} Find its domain
{–2, 4, 8}
Find its range
{–5, 7, 9}

6. Function
Function
A function is a set of ordered pairs (a relation) in which to each first component, there corresponds exactly one second component.
Is this a function?
{(1, 2), (2, 4), (3, 6), (4, 8), (, 10)}
Yes
{(2, 4), (3, 9), (4, 16), (5, 25), (6, 36)}
{(36, 6), (25, 5), (25, -5), (16, 4), (9, 3)} No
{(1, 6), (2, 5), (3, -5), (4, 4), (4, 3)}

7. Y as Function ox X Given the equation: y = 3x – 4
Since the equation describes a set of ordered pairs, it describes a relation.
Since, for each value of the domain, there is only one value of the range, the equation also describes a function.

8. Y as Function of X y is a Function of x
Any equation in x and y where each value of x determines exactly one value of y is called a function.
In this case, we say that y is a function of x.
y = 3x – 4
Variable y is a function of x.
Variable x is the independent variable.
Variable y is the dependent variable.

9. Function
Comments
Function is always a relation, but relation is not always a function.
E.g., {(2, 1), (2, 3)} is a relation, but not a function
Another view of a Function x (independent y (dependent variable) variable)
y = 3x - 4 (function)

10. Example
Determine whether the equations define y to be a function of x.
y = x2
x = y2

11. Solution
y = x2
The table shows that for each value of x, there is exactly one value of y. So, the relation is a function.

12. Solution
x = y2
We construct a table of ordered pairs for the equation x = y2. Because y is squared, it will be more convenient to substitute values for y and compute the corresponding values for x.
The table shows that for each value of x, there are more than one value of y. Thus, the equation does not describe a function.

13. Function Notation Write y = 2x – 3 as f(x) = 2x – 3.
Read this as: “f of x = 2x – 3.”
The notation y = f(x), read “y equals f of x,” means that the variable y depends on the value of x.

14. Examples Let f (x) = 2x – 3. Find the following f (3) f (–1) f (0)
the value of x for which f (x) = 5.
f(x) = 2x = 2x – 3 8 = 2x x = 4

15. Graph of a Function Given: f (x) = | x | Graph the function
Determine the domain and range

16. Domain and Range Domain—set of all real numbers
D = {x | x is a real number}
Range—set of all positive numbers and 0
R = {y | y ≥ 0}

17. Another View of Functionx y
Relation x y 2 3
0 1
4 9
Function

18. Determine whether a graph represents a function
A vertical line test can be used to determine whether the graph of an equation represents a function.
If any vertical line intersects a graph more than once, the graph cannot represent a function, because to one number x, there would correspond more than one value of y.
The graph in Figure represents a function, because every vertical line that intersects the graph does so exactly once.

19. Determine whether a graph represents a function
The graph in Figure does not represent a function, because some vertical lines intersect the graph more than once.

20. Determine whether a graph represents a function
Determine whether each graph represents a function.

21. Excel Graphing
MS Excel Graphing
functions.xls