1. Learning Objectives for Section 2.1 Functions
MAT SPRING 2009
Learning Objectives for Section 2.1 Functions
You will be able to give and apply the definition of a function.
You will be able to identify domain and range of a function.
You will be able to use function notation.

2. Function Definition Definition of Function:
MAT SPRING 2009
Function Definition
Definition of Function:
A function is a correspondence between two sets D and R such that each element of the first set, D, maps to EXACTLY one element of the second set, R.
The first set (inputs) is called the __________________, and the set of corresponding elements in the second set (outputs) is called the ___________________.

3. MAT SPRING 2009
Function Diagram
You can visualize a function by the following diagram which shows a correspondence between two sets:
D, the domain of the function, gives the diameter of pizzas, and
R, the range of the function gives the cost of the pizza. 10 12 16
9.00
12.00
10.00
domain D
range R

4. Examples Determine if the following relations are functions. 1) 2) 3
1) ) 3 -2 8 1 2 3 1 5 6

5. Examples
Determine if the following relations are functions.

6. Functions Specified by Equations
MAT SPRING 2009
Functions Specified by Equations
Consider the equation
Input: x = -2 -2
Input
Output
Process: square (–2), then subtract 2
(-2, 2) is an ordered pair of the function. 2
Output: result is 2

7. Functions as Equations
To determine if an equation represents y as a function of x:
You should be able to solve for y, using only one equation, so that each input (x) has only one output (y).
Note: If y is raised to an even power or if y is in the absolute value, then the equation does NOT represent y as a function of x.

8. Examples
These are functions: These are NOT functions:

9. Vertical Line Test for a Function
MAT SPRING 2009
Vertical Line Test for a Function
If you have the graph of an equation, there is an easy way to determine if it is the graph of a function.
It is called the ________________ ___________ ___________
An equation defines a function if each vertical line in the coordinate system passes through AT MOST ONE POINT on the graph of the equation.
IF ANY VERTICAL LINE PASSES THROUGH TWO OR MORE POINTS ON THE GRAPH, THEN THE EQUATION DOES NOT DEFINE A FUNCTION.

10. Vertical Line Test for a Function (continued)
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Vertical Line Test for a Function (continued)
This graph is not the graph of a function because you can draw a vertical line which crosses it twice.
This is the graph of a function because any vertical line crosses only once.

11. MAT SPRING 2009
Functional Notation
FUNCTIONAL NOTATION is used to describe functions. The variable y will now be called f (x).
f (x) is read as “ f of x” and simply means the y coordinate of the function corresponding to a given x value.
Our previous equation can now be expressed as

12. Evaluating Functions Consider our function What does f (-3) mean?
MAT SPRING 2009
Evaluating Functions
Consider our function
What does f (-3) mean?

13. MAT SPRING 2009
Some Examples
Given , complete the following.

14. Domain of a Function Consider
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Domain of a Function
Consider
Question: For what values of x is the function defined?

15. Domain of a Function (continued)
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Domain of a Function (continued)
To find the domain of a square root function:
Set the radicand (expression under the radical) ≥ 0 and solve for x.
Answer: To find the domain of

16. Domain of a Function (continued)
MAT SPRING 2009
Domain of a Function (continued)
Example: Find the domain of the function

17. Domain of a Function: A Rational Example
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Domain of a Function: A Rational Example
Find the domain of

18. Domain of a Rational Function
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Domain of a Rational Function
Remember, if the denominator is equal to zero, the function is _________________________!
To find the domain of a rational function (variable in the denominator):
Set the denominator equal to zero and solve for x to determine which x values must be EXCLUDED.
The domain will be all real numbers EXCEPT for these values.

19. Domain of a Function: Another Example
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Domain of a Function: Another Example
Find the domain of

20. Domain of a Function: And Another Example
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Domain of a Function: And Another Example
Find the domain of

21. MAT SPRING 2009
Domain of a Function
The domain of any unrestricted linear, quadratic, cubic function, or absolute value function is the set of ____________ __________________.
For example, the following functions have a domain of
Check these out on your graphing calculator.

23. MAT SPRING 2009
Range of a Function
The range of a function is the set of all outputs (y-values) .
To find the range of a function, look at its graph to determine the y-values that are included on the graph.
Check if your graph has a maximum or minimum y-value to help you determine the range.

24. Example: State the range of g using interval notation.
MAT SPRING 2009
Range of a Function
Example: State the range of g using interval notation.