8.8 Function Notation and Linear Functions
Learning Objectives
- I can convert an equation to function notation.
- I can input a value into a function in order to get an output value.
Introduction
Suppose you just purchased a used car, and the number of miles on the odometer can be represented by the equation , where is the number of miles on the odometer, and is the number of miles you have driven it. Could you convert this equation to function notation? How many miles will be on the odometer if you drive the car 700 miles? In this concept, you'll learn how to convert equations such as this one to function notation and how to input a value into a function in order to get an output value.
Guided Learning
So far, the term function has been used to describe many of the equations we have been working with. The concept of a function is extremely important in mathematics. Not all equations are functions. To be a function, for each value of there is one and only one value for .
A function is a relationship between two variables such that the input value has ONLY one unique output value.
Recall from a previous Concept that a function rule replaces the variable with its function name, usually . Remember that these parentheses do not mean multiplication. They separate the function name from the independent variable, .
is read “the function of ” or simply “ of .”
If the function looks like this: , it would be read of equals 3 times minus 1.
Using Function Notation
Function notation allows you to easily see the input value for the independent variable inside the parentheses.
Example A
Consider the function .
Evaluate .
Solution: The value inside the parentheses is the value of the variable . Use the Substitution Property to evaluate the function for .
To use function notation, the equation must be written in terms of . This means that the variable must be isolated on one side of the equal sign.
Example B
Rewrite using function notation.
Solution: The goal is to rearrange this equation so the equation looks like . Then replace with .
Functions as Machines
You can think of a function as a machine. You start with an input (some value), the machine performs the operations (it does the work), and your output is the answer. For example, takes some number, , multiplies it by 3 and adds 2. As a machine, it would look like this:
When you use the function machine to evaluate , the solution is .
Example C
A function is defined as . Determine the following:
a)
b)
Solution:
a) Substitute into the function .
b) Substitute into the function .
Example D
Functions have only one output value for each input value. When looking at graphs, that means that there should be only one x value for each y value. Look at the graphs to decide if they show a function.
This graph is not a function. There is more than one value for the y values.
This graph is a function. There is only one value for each x and y value.
Guided Practice
Rewrite the equation in function notation where , and then evaluate , and .
Stop and check your work with a partner.
Solution
First we need to solve for .
Adding to both sides gives , and dividing by 2 gives
Now we just replace the with to get .
Now we can evaluate for , and :
Review
- Use function notation to solve equations.
Function
A function is a relationship between two variables such that the input value has ONLY one unique output value.