1. 1.4: Functions Objectives: To determine if a relation is a function
To find the domain and range of a function
To evaluate functions

2. Vocabulary
As a class, use your vast mathematical knowledge to define each of these words without the aid of your textbook.
Relation
Function
Input
Output
Domain
Range
Set-Builder Notation
Interval Notation
Function Notation

3. Relations
A mathematical relation is the pairing up (mapping) of inputs and outputs.

4. Relations
A mathematical relation is the pairing up (mapping) of inputs and outputs.
Domain: the set of all input values
Range: the set of all output values

5. Calvin and Hobbes!
A toaster is an example of a function. You put in bread, the toaster performs a toasting function, and out pops toasted bread.

6. Calvin and Hobbes! “What comes out of a toaster?”
“It depends on what you put in.”
You can’t input bread and expect a waffle!

7. What’s Your Function?
A function is a relation in which each input has exactly one output.
A function is a dependent relation
Output depends on the input
Relations
Functions

8. What’s Your Function?
A function is a relation in which each input has exactly one output.
Each output does not necessarily have only one input
Relations
Functions

9. How Many Girlfriends?
If you think of the inputs as boys and the output as girls, then a function occurs when each boy has only one girlfriend. Otherwise the boy gets in BIG trouble.
Darth Vadar as a “Procurer.”

10. What’s a Function Look Like?

11. What’s a Function Look Like?

12. What’s a Function Look Like?

13. What’s a Function Look Like?

14. Exercise 1a
Tell whether or not each table represents a function. Give the domain and range of each relationship.

15. Exercise 1b
The size of a set is called its cardinality. What must be true about the cardinalities of the domain and range of any function?

16. Exercise 2 Which sets of ordered pairs represent functions?
{(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)}
{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}
{(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)}
{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}

17. Exercise 3
Which of the following graphs represent functions? What is an easy way to tell that each input has only one output?

18. Vertical Line Test
A relation is a function iff no vertical line intersects the graph of the relation at more than one point
If it does, then an input has more than one output.
Function
Not a Function

19. Functions as Equations
To determine if an equation represents a function, try solving the thing for y.
Make sure that there is only one value of y for every value of x.

20. Exercise 4
Determine whether each equation represents y as a function of x.
x2 +2y = 4
(x + 3)2 + (y – 5)2 = 36

21. Set-Builder Notation
Since the domain or range of a function is often an infinite set of values, it is often convenient to represent your answers in set-builder notation.
Examples:
{x | x < -2} reads “the set of all x such that x is less than negative 2”.

22. Set-Builder Notation
Since the domain or range of a function is often an infinite set of values, it is often convenient to represent your answers in set-builder notation.
Examples:
{x : x < -2} reads “the set of all x such that x is less than negative 2”.

23. Interval Notation
Another way to describe an infinite set of numbers is with interval notation.
Parenthesis indicate that first or last number is not in the set:
Example: (-, -2) means the same thing as x < -2
Neither the negative infinity or the negative 2 are included in the interval
Always write the smaller number, bigger number

24. Interval Notation
Another way to describe an infinite set of numbers is with interval notation.
Brackets indicate that first or last number is in the set:
Example: (-, -2] means the same thing as x  -2
Infinity (positive or negative) never gets a bracket
Always write the smaller number, bigger number

25. Domain and Range: Graphs
Domain: All x-values (L → R)
{x: -∞ < x < ∞}
Range: All y-values (D ↑ U)
{y: y ≥ -4}
Range: Greater than or equal to -4
Domain: All real numbers

26. Exercise 5
Determine the domain and range of each function.

27. Domain and Range: Equations
Domain: What you are allowed to plug in for x.
Easier to ask what you can’t plug in for x.
Limited by division by zero or negative even roots
Can be explicit or implied
Range: What you can get out for y using the domain.
Easier to ask what you can’t get for y.

28. Exercise 6a
Determine the domain of each function.
y = x2 + 2

29. Exercise 6b
Determine the domain of each function.

30. Dependent Quantities
Functions can also be thought of as dependent relationships. In a function, the value of the output depends on the value of the input.
Independent quantity: Input values, x-values, domain
Dependent quantity: Output value, which depends on the input value, y-values, range

31. Exercise 7
The number of pretzels, p, that can be packaged in a box with a volume of V cubic units is given by the equation p = 45V In this relationship, which is the dependent variable?

32. Function Notation
In an equation, the dependent variable is usually represented as f (x).
Read “f of x”
f = name of function; x = independent variable
Takes place of y: y = f (x)
f (x) does NOT mean multiplication!
f (3) means “the function evaluated at 3” where you plug 3 in for x.

33. Exercise 8 Evaluate each function when x = -3. f (x) = -2x3 + 5
g (x) = 12 – 8x

34. Exercise 9 Let g(x) = -x2 + 4x + 1. Find each function value. g(2)
g(t)
g(t + 2)

35. Assignment: Continue Pgs 118-119 #47-79 odd
1.4: Functions
Objectives:
To determine if a relation is a function
To find the domain and range of a function
To evaluate functions
Assignment: Continue Pgs #47-79 odd