1. An Introduction to Functions
Linear Algebra

2. Why do we need to learn about functions?
Functions are rules.
Functions allow us to make predictions.
Functions can allow us to classify the data in our environment.

3. Define-
A relation is any relationship between a set of inputs and a set of outputs
A function is more specific.
A function is a rule that relates two quantities so that each input value corresponds to exactly one output value.
To be a function, every x value will have only one corresponding y value

4. (x,y) The input is first coordinate or x
The output is second coordinate or y
(x,y)
This table represents the ordered pairs
(-5,10) (0,10) and (5,10)
The domain of a function is the set of all inputs Xs
The range of a function is the set of all outputs Ys
(think list the x values)
Domain {-5,0,5}
Range {10,0,10}
(think list the y values)

5. Function or Non-Functions
You can tell if a relation is a function from any of it’s forms:
A Table
A Set of Ordered Pairs
A Graph
Mapping
An Equation
A Verbal Model

6. Determine if the following relations are functions:
FROM A TABLE
No x can have more than one different y
No repeat xs allowed
Yes.
Each “x” value has only one “y” value.
No.
The “x” value 4 has two different “y” values.

7. Determine if the following relations are functions:
FROM A Set of Ordered Pairs
No x can have more than one different y
No repeat xs allowed
{(2, 5), (3, 9), (-4, 10), (6, -6), (-7, 5)}
{(3, 7), (-8, 2), (-1, 0), (3, 8), (2, 6)}
Yes.
Each “x” value has only one “y” value.
No.
The “x” value 3 has two different “y” values.

8. Determine if the following relations are functions:
FROM A Mapping
No x can have more than one different y
No repeat xs allowed
Yes.
Each “x” value has only one “y” value.
No.
The “x” value 85 and 95 has two different “y” values.

9. Determine if the following relations are functions:
FROM A Graph
No x can have more than one different y
Use the “Vertical Line Test”

10. In order for a graph to be a function, each x can only have ONE y.
Give an example of why this graph is not a function.
Is this graph a function? Why or why not?
Yes, each x input has only one y output.

11. Is this graph a function? Why or why not?
NO, each x input has more than one y output.
Is this graph a function? Why or why not?
Yes, each x input has only one y output.

12. Use the Vertical Line Test
From a Graph
Use the Vertical Line Test
When looking at a graph, you can tell if a drawing is a function if it passes the vertical line test.
This means you can draw a vertical line and it will only touch the drawing (graphed figure) one time.
If it touches the drawing (graphed figure) more than once, it is not a function.

13. Determine if the relationship represents a function
Determine if the relationship represents a function. Does it pass the vertical line test? NO
The relationship is not a function.

14. The relationship is a function.
Determine if the relationship represents a function. Does it pass the vertical line test? x y
YES
The relationship is a function.

15. Determine if the relationship represents a function
Determine if the relationship represents a function. Does it pass the vertical line test?
YES
The relationship is a function.