Function Definition and Notation – Guided Notes
Definition of Function
You have already investigated linear relationships between two variables. Now we are going to look
at a specific type of relation known as a function, a very important concept in mathematics.
Simple Definition – Version 1 - Function: A function is a rule that describes how to modify an input
to create a specific output.
The definition of function can also be remembered as a diagram. This type of diagram will be
called a machine diagram of a function, and it says the same thing as the word definition above.
INPUT
RULE
OUTPUT
something
happens
Example: The rule "add 3 to the input" is a function.
The function can take any input (like “1”) perform its rule ("add 3 to the input") and create an
output (“4”). We can perform the rule on any input. If the input is 1, 3, 5, or 20, then the
output would be 4, 6, 8, or 23, respectively. This is represented on the machine diagram below.
To represent "any input" mathematicians often use a variable, like "x." The output of this rule
could then be described by the expression "x + 3."
Question: If we graphed the input-output pairs from above, what would it look like? Something we
know? How do you know?
Function Definition and Notation – Guided Notes
Function Notation
The notation for functions is often given like this: the symbol for the input is x, and the symbol for the
output is f(x). Here is a machine diagram with this notation:
With this notation, it’s possible to write an input-output formula as an equation.
Important note: Remember, x is just a variable. f(x) = x+3 could also be written f(y) = y +3 or
f(★) = ★ + 3.
Example 1: f(x) = 2x + 1 means:
“The rule is: ______________________________________ “
We can also say, "When the input is x, the output is ___________________”
Example 2: f(3) = 7 means: ____________________________________________
Finding the output for a given input is called evaluating the function. Using the f(x) notation,
the easiest way to evaluate is to substitute the input number in place of x.
Example 3: For the function f(x) = 2x + 1, when the input is 3, what is the output?
Vocabulary Note:
We often use “__________________________ variable” to describe the input, and
“______________________ variable” to describe the output.
You can remember this by thinking that the value of our output, f(x), depends on the value we input.
For example, if we took a train ride where the price of the ticket depended on how many miles we
traveled; the number of miles would be the independent variable – or the ___________ to the function,
and the price of the ticket the dependent variable, or the ______________ of the function.
Function Definition and Notation – Guided Notes
Group Exploration Exercises
1. Answer these questions about the function f(x) =
x
.
3
a. Write a word description of the function rule. (What is done to the input to get the output?)
b. When the input is 12, what is the output?
c. Find the value of f(15).
d. Evaluate f(30).
e. Is it true that f(6) = 18 ? Why or why not.
2. Answer these questions about the function f (x) = x2 + 1.
a. When the input is –7, what is the output?
b. Evaluate f(6).
c. Find f(–6).
d. Find an input number that would result in an output of 5.
e. Can you find another input number that would also result in an output of 5?
Function Definition and Notation – Guided Notes
3. Here is an input-output table for a function f(x).
Independ.
Variable
x
Dependent
Variable
f(x)
0
1
2
3
4
5
6
7
8
5
0
4
9
–1
3
2
Remember that each input-output pair can be expressed using function notation.
For example, the first row of the table says that f(0) = 8.
a. Using the second and third rows of the table, fill in the blanks in these statements.
f(____) = _____
f(____) = _____
b. What is the value of f(6) ?
c. Which value of x would result in a negative number output?
d. Find an input number x that would make f(x) = 2.
e. One of your friends thinks that f(4) = 3.
Another one of your friends thinks that f(4) = 9.
Which of these is correct?
Explain in a way that the incorrect friend could understand his/her mistake.