Imagine every student in this room is assigned their birth month.
Does every student have a month? Yes. Does any student have two different birth months? No.
This is a function.
In middle school, you learned that a function is a rule that assigns each input to exactly one output.
Each input has a single, predictable result.
The Domain is the complete set of all allowed inputs.
Think of the domain as the "source" or the "starting point." If a value isn't in the domain, the function doesn't know what to do with it!
In our birthday example: The domain is {all students in this room}.
The Range is the complete set of all outputs produced by the function.
The range isn't just "any number"—it's the specific collection of values the function actually "hits."
In our birthday example: The range is {months that are actually birthdays of students}.
A function from the domain to the range is a rule that assigns to each element of the domain exactly one element of the range.
Notice that multiple inputs can share an output (like two students having a March birthday). That is still a function!
Tables are great, but they are bulky. We need a shorthand.
Instead of saying "When the input is 5, the output is 13," we want a mathematical sentence.
Function notation gives us that language. It is the universal language of higher mathematics.
When you see , read it aloud as:
NEVER say "f times x."
The parentheses here are not for multiplication; they are a "holder" for the input value.
In , the parentheses mean multiply. In , the parentheses mean input.
If , then is . It is NOT . This is the single most common mistake in Algebra 1!
Let . Find .
Result: Translation: "The output is 13 when the input is 5."
Result: Note: Parentheses are vital when plugging in negative numbers!
They are not the same thing!
You "use" the function to "find" the value .
The machine follows a rule. You drop in, and pops out. The machine doesn't change; only the parts you feed it do.
Functions aren't always named .
Choose names that help you remember what the numbers represent!
In function notation, we write: , , and .
A function assigns to each student their height in inches.
If , find .
Work it out: .
We've mastered the notation and the definitions.
Next, we will look at how these functions look on a coordinate plane and how the Domain can be restricted by the real world.
Click to begin the narrated lesson
Understand function definition