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Understand Function Definition

Formalizing the Relationship

Learning Objectives for This Deck

  • Define a function using formal set language.
  • Identify the domain and range of a relationship.
  • Master function notation and interpretation.
  • Distinguish between a function and its output.

Hook: Every Student Has a Birth Month

Imagine every student in this room is assigned their birth month.

  • Inputs: The students in the class.
  • Outputs: The 12 months of the year.

Does every student have a month? Yes. Does any student have two different birth months? No.

This is a function.

Grade 8 Recap: Inputs and Outputs

In middle school, you learned that a function assigns each input exactly one output.

Input (x) Output (y)
1 5
2 10
3 15

Each input has a single, predictable result.

The Domain: Set of All Valid Inputs

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: Set of All Actual Outputs

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}.

The Formal Definition of a Function

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.

  1. Each: No input can be left behind.
  2. Exactly One: No input can have two different "answers."

Mapping a Function: Many-to-One Is Fine

A mapping diagram showing domain set with elements A, B, C mapped to range set with elements 1, 2, 3. B and C both map to 3.

Why We Need Function Notation

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.

Breaking Down the Parts of

  • : The name of the function (the "Machine").
  • : The input variable (the "Raw Material").
  • : The rule (the "Instructions").
  • : The output value (the "Finished Product").

Reading Aloud: Never Say "Times"

When you see , read it aloud as:

"f of x"

NEVER say "f times x."

The parentheses here are not for multiplication; they are a "holder" for the input value.

Watch Out: The Multiplication Trap

STOP and read carefully:

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!

Evaluation Example: Step-by-Step Linear Function

Let . Find .

  1. Start with the rule:
  2. Substitute the input:
  3. Calculate:

Result:
Translation: "The output is 13 when the input is 5."

Evaluation Example: Quadratic with Negative Input

Let . Find .

  1. Start with the rule:
  2. Substitute the input:
  3. Calculate:

Result:
Note: Parentheses are vital when substituting negative numbers!

Distinguishing the Function from Its Output

They are not the same thing!

  • is the Function. It is the entire machine, the blueprint, the rule.
  • is the Output. It is a specific number, a result, a coordinate.

You "use" the function to "find" the value .

Input In, Output Out: The Function Machine

A machine diagram with an input slot at top labeled x and an output tray at bottom labeled f(x). The box in the middle is labeled f.

Functions Can Have Any Name

Functions aren't always named .

  • (often used for a second function)
  • (used when input is time)
  • (used when output is cost)

Choose names that help you remember what the numbers represent!

Translating a Table into Function Notation

x output
1 5
2 7
3 9

In function notation, we write:
, , and .

Upgrading from Grade 8 to High School Language

Informal (8th Grade) Formal (High School)
Input Domain Element ()
Output Range Element ()
Rule Function ()
y-coordinate

Knowledge Check: Domain and Range

A function assigns to each student their height in inches.

  1. What is the domain?
  2. Is it possible for two different students to have the same height?
  3. If so, is it still a function?

Knowledge Check: Evaluating

If , find .

  • A) 4
  • B) 16
  • C) 28
  • D) 4

Work it out: .

Next Steps: Seeing Functions on a Graph

We've mastered the notation and the definitions.

Next, we will look at how these functions appear on a coordinate plane and how the Domain can be restricted by the real world.