3.2: Function Notation
What is Function Notation?
Objectives
The lesson objective for Function Notation is:
 Understanding function notation.
Introduction
If you think of the relationship between two quantities, you can think of this relationship in terms of a ‘function machine’. The function machine shows how the function responds to various functions. If I triple the input and subtract one, the machine will respond to the function in such a way that it will convert into . If the function is , and 3 is fed into the machine, comes out.
Watch This
Khan Academy Functions as Graphs
Guidance
To represent functions as equations, the symbol is often used. The symbol is pronounced as “ of .” This means that the equation is a function that is written in the terms of the variable . An example of such a function is . Functions can also be written using a letter other than and a variable other than . An example of such a case is and . In addition to representing functions as equations, you have been shown a variety of ways by which a function can be represented. These ways included:
 As a graph
 As ordered pairs
 As a table of values
 As an arrow or mapping diagram
 As mapping notation
When a function is represented as an equation, an ordered pair can be determined by evaluating various values of the assigned variable. If , then
Graphically, if , this means that the point (4, 8) is a point on the graph of the line.
Example A
If find.
a)
b)
c)
To determine the value of the function for the assigned values of the variable, substitute the values into the function.
Example B
Functions can also be represented as mapping rules. If find the following in simplest form:
a)
b)
c)
Example C
If
a) evaluate
i)
ii)
iii)
b) Find a value of ‘’ where does not exist.
c) Find in simplest form
d) Find ‘’ if
a)
b) The function will not exist if the denominator equals zero.
Therefore, if , then does not exist.
c)
d)
Example D
Functions can also be used to represent and to solve word problems. The following is an example of this application.
The value of a digital camera years after it was bought is represented by the function
a) Determine the value of and explain what the solution mean to this problem.
b) Determine the value of then and explain what this represents.
c) What was the original cost of the digital camera?
a)
This means that the camera is valued at $675, 4 years after it was purchased.
b)
The digital camera has a value of $525, 7 years after it was purchased.
c)
The original cost of the camera was $875.
Vocabulary
 Function
 A function is a set of ordered pairs that shows a relationship where there is only one output for every input. In other words, for every value of , there is only one value for .
Guided Practice
1. If find:
i)
ii)
2. If find ‘’ if
3. The emergency brake cable in a truck parked on a steep hill breaks and the truck rolls down the hill. The distance in feet, , which the truck rolls, is represented by the function .
i) How far will the truck roll after 9 seconds?
ii) How long will it take the truck to hit a tree which is at the bottom of the hill 600 feet away?
Round your answer to the nearest second.
Answers
1.
i)
ii)
2.
3.
i)
After 9 seconds, the truck will roll 40.5 feet.
ii)
The truck will hit the tree in approximately 35 seconds.
Summary
Using function notation to represent a function is similar to expressing the function as a formula. The formula is used to determine the value of a function for a given value. The formula can also be used to calculate the value of the variable when a value is given for the entire function. These concepts can then be applied to real life situations to determine the value of an indicated variable.
Problem Set

If , find expressions for the following:

If , determine the value of ‘’ when:
Solve the following problem:
The value of a Bobby Orr rookie card years after its purchase is
i) Determine the value of and explain what the solution means.
ii) Determine the value of when and explain what this represents.
iii) Determine the original price of the card.
Answers
 If , find... (a) (c) (e)
 If determine... (a) (c) (e)
Solve the following...
i)
The solution represents the value of the card six years after it was purchased. The value of the card is $688.00.
ii)
. This represents the value of the card 8 years after it was purchased.
iii)
The original purchase price of the card was $520.00.
Summary
In this chapter you learned about linear functions. You learned what is meant by a function, function notation and how to identify a function from a graph. You also learned how to recognize whether or not a graph represented a function or a relation by doing the vertical line test. Using function notation as a formula, you learned to determine the value for an indicated variable and how to evaluate a function for a given value.