Section 1.1 Functions and Function Notation

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Section 1.1Functions and Function Notation
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What is a function?

• When you hear the term function what do you think
  of?
• Lets get some examples of what a function is

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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

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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

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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?

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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)

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Table Representation
Month 1 2 3 4 5 6
Milk 1.65 1.72 1.78 1.81 1.75 1.72
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

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

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Which one is not a function?
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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?

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• 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.