Understanding the difference between a function and a relation is important for your pre-calculus studies!

What is a Relation?

• A relation is just a group of pairs of input and output values.
• You can think of it as a collection of points that connect one value to another.
• For example, let’s look at these pairs: $(1, 2)$, $(2, 3)$, and $(1, 4)$.
• In this case, the first number is the input and the second number is the output.
• The first number (input) can be linked to different second numbers (outputs).

What is a Function?

• A function is a special kind of relation.
• In a function, every input is linked to exactly one output.
• This means if you have an input of 1, it can only give you one output.
• Going back to our earlier pairs, if you have $(1, 2)$ and $(1, 4)$, this can't be a function.
• That’s because the input 1 is pointing to two different outputs.

Visualizing Functions

• A simple way to see the difference is by using a mapping diagram:
  • In a relation, you might see one input connecting to several outputs.
  • In a function, every input has just one arrow going to one output.

Why This Matters

• Functions are very important because they make sure that you get the same output for the same input.
• This consistency helps you understand equations and graphs better.

So, when you're learning about these topics, remember this: all functions are relations, but not all relations are functions!