1. Section 1.1 Functions and Function Notation

2. What is a function?
When you hear the term function what do you think of?
Let’s get some examples of what a function is

3. Definition of function
A function is a rule that assigns a unique output to each input
We say that the output is a function of the input
The inputs and outputs are often represented by variables
A function can be described using words, data in a table, points on a graph, or a formula

4. What do we mean by unique?
Each input has one (and only one) output
An output can have more than one input
Example: the squaring function
Takes an input and gives its square as an output
For example inputs 2 and -2 have an output of 4
Therefore they each have a unique output

5. What about these variables?
We can refer to the input variable as the independent variable and the output variable as the dependent variable
Do these variables have to represent numbers?
Can you give an example of a function where the variables do not represent numbers?

6. Function notation Example: E = mc2
In general, to indicate that Q is a function of a quantity t we can say
Q is a function of t
Q equals f(t)
Q = f(t)
Which is the independent variable? Dependent?
Dependent = f(Independent)
Output = f(Input)

7. Table Representation
Month 1 2 3 4 5 6
Milk
$1.65
$1.72
$1.78
$1.81
$1.75
Gas
$2.13
$2.16
$2.26
$2.29
$2.41
$2.48
Which of the three variables are functions of one of the other variables?
Using function notation, show what is going on in month 3 and determine what is the input and the output
f(3) = 1.78
g(3) = 2.26

8. A Function!!! Not a Function!!! Month (Input) Milk (Output) 1 $1.65 2
$1.72 3
$1.78 4
$1.81 5
$1.75 6
Milk
(Input)
Gas
(Output)
$1.65
$2.13
$1.72
$2.16
$1.78
$2.26
$1.81
$2.29
$1.75
$2.41
$2.48

9. Graphical Representation
The following graph represents the height of a ball thrown straight up in the air as a function of time traveled
Note: Do not confuse the graph with the actual motion of the ball

10. The Vertical Line Test
If there is a vertical line which intersects a graph in more than one point, then the graph does not represent a function

11. Which one is not a function?

12. Symbolic Representation
Recall this graph
The symbolic representation for the height of the ball (in feet) as a function of time (in seconds) is
What is the independent variable? Dependent?
What is h(1) and what does it represent?

13. One last problem
The sales tax on an item is 7%. Express the total cost, C, in terms of the price of the item, P.