This document provides definitions and examples related to functions in mathematics. It defines a function as a rule that assigns each input value to exactly one output value. A function can be described using an equation. The domain of a function refers to the input or x-values, while the range refers to the output or y-values. A relation represents a function if it passes the vertical line test, meaning a vertical line can intersect the graph at only one point as it moves from left to right. Examples demonstrate how to determine the domain of functions by identifying restrictions like imaginary or zero denominators.

1. Vocabulary
• Function: Rule that assigns each input value
to exactly one output.
Usually we describe a function with an equation.

2. Grading Function

4. Vocabulary
• Function: Rule that assigns each input value
to exactly one output.
Usually we describe a function with an equation.
• Domain: x –values, or input values.
• Range: y – values , or output values.

5. How do we know if a graph is a
function?
• If it passes THE VERTICAL LINE TEST
The vertical line test
1.Move a vertical line across the entire graph
from left to right.
2.If it intersects the graph only ONCE in all
locations, then it is a function.

6. Is this a function?

7. Is this a function?

8. Is this a function?

9. What is the Domain and Range?

10. What is the Domain and Range?

11. What is the Domain and Range?

12. Function notation
)(xf

13. Practice
1. Is a function represented? If so what is the
domain and range.
a. (2,-4) (4,4) (6,-1) (5,0) (1,6) (2,4) (3, -2)
b.
2. If , then find f (0) , f (-2) , and f (3a)5)( 2
−= xxf

14. Finding the domain from the equation
Example: What is the domain of the function given
by
Functions only have real number outputs.
Anything that gives an imaginary number or a
zero denominator is a restriction.
How to find the domain:
Assume that the domain is the set of all real
numbers except for the restrictions.
5)( −= xxf

15. Examples
• What is the domain?
• What is the domain?
7
2
)(
+
=
x
xf
9
7
)( 2
−
=
x
xf

16. More challenging examples
• What is the domain?
• What is the domain?
4
32
)(
+
−=
xx
xf
6
9
)(
+
−
=
x
x
xf

17. Practice
• What is the domain?
• What is the domain?
• What is the domain?
xx
xf
5
5
)( 2
−
=
7)( 2
−= xxf
3
7)( −= xxf