Mathematics · 9th Grade · The Language of Algebra · Weeks 1-9

Introduction to Functions

Defining functions, identifying domain and range, and distinguishing functions from relations.

• Common Core State Standards
• CCSS.Math.Content.HSF.IF.A.1
• CCSS.Math.Content.HSF.IF.A.2

About This Topic

The introduction to functions marks a conceptual shift in 9th grade mathematics: from working with specific numbers and operations to thinking about relationships between quantities. In the U.S. Common Core framework, this topic formally defines what makes a relation a function, focuses on domain and range, and introduces the vertical line test as a graphical tool. For many students, this is their first sustained encounter with the idea that math can describe how one quantity depends on another.

Understanding domain and range requires students to think about a set of inputs and a corresponding set of outputs as objects in their own right, not just individual pairs. This kind of set-based thinking is new for most 9th graders and benefits from multiple representations: tables, graphs, mapping diagrams, and equations all present the same underlying relationship differently, and fluency with all four builds a robust understanding of functions.

Active learning activities that ask students to create, test, and critique function examples are particularly effective here. When students must produce their own mappings and test whether they are functions, they internalize the definition more deeply than when they only classify teacher-provided examples.

Key Questions

1. Differentiate between a relation and a function using various representations.
2. Explain how to determine the domain and range of a function from a graph or equation.
3. Justify the importance of the vertical line test in identifying functions.

Learning Objectives

• Classify a given relation as a function or not a function, justifying the classification using the definition of a function.
• Determine the domain and range of a function represented by a graph, equation, or table of values.
• Compare and contrast relations and functions, identifying key characteristics of each.
• Create a mapping diagram that represents a function and another that represents a relation, explaining the difference.
• Evaluate the validity of a relation as a function using the vertical line test on its graph.

Before You Start

Introduction to Coordinate Planes

Why: Students need to be able to plot and interpret points on a coordinate plane to understand graphical representations of relations and functions.

Sets and Set Notation

Why: Understanding how to define and work with sets of numbers is fundamental to grasping the concepts of domain and range.

Basic Equation Solving

Why: Students must be able to solve simple equations to find output values for given input values when working with function notation.

Key Vocabulary

Relation	A set of ordered pairs, where each first element (input) is related to each second element (output) in some way.
Function	A special type of relation where each input has exactly one output.
Domain	The set of all possible input values (x-values) for a relation or function.
Range	The set of all possible output values (y-values) for a relation or function.
Ordered Pair	A pair of numbers written in a specific order, typically (x, y), representing a point on a graph or a specific input-output relationship.

Watch Out for These Misconceptions

Common Misconception

Students think a function cannot have the same output for two different inputs (i.e., they confuse 'function' with 'one-to-one function').

What to Teach Instead

Clarify that the restriction is on inputs, not outputs: each input maps to exactly one output, but two different inputs may share the same output. Use a temperature example: 75 degrees on Monday and 75 degrees on Friday are fine, since each day (input) has one temperature (output).

Common Misconception

Students assume all equations are automatically functions.

What to Teach Instead

Show the equation x = y squared, which fails the vertical line test. Ask students to find two y-values for a single x-value to confirm it is not a function. Graphing and the vertical line test must always be used as a check.

Active Learning Ideas

Mapping Diagram Gallery: Function or Not?

Post 10 mapping diagrams around the room, some representing functions and some representing relations that are not functions. Students circulate with sticky notes, marking each as 'function' or 'not a function' and writing one justification sentence. The class then reviews each station's results and resolves disagreements.

30 min·Individual

Think-Pair-Share: Design a Function

Ask students individually to draw a mapping diagram with exactly 4 inputs that is a function, then revise it to make it not a function by changing only one arrow. Partners compare their diagrams and explain what change violated the function definition. Pairs share the most interesting cases with the class.

20 min·Pairs

Vertical Line Test Station Rotation

Set up four stations with different graphs (linear, parabola, circle, scattered points). Students apply the vertical line test at each station using a piece of string or a ruler as a physical vertical line, record the domain and range visually, and determine whether the graph represents a function. Groups discuss any cases they disagree on.

30 min·Small Groups

Real-World Connections

• A car's odometer reading is a function of time; for any given time, there is only one odometer reading. This helps track mileage for maintenance schedules.
• A thermostat setting is a function of desired temperature; for each desired temperature, the thermostat sends a specific signal to the heating or cooling system.
• The price of a stock at closing each day is a function of time; each day has a single closing price, which is crucial for financial analysis and investment tracking.

Assessment Ideas

Quick Check

Provide students with 3-4 sets of ordered pairs. Ask them to write 'Function' or 'Not a Function' next to each set and briefly explain why. For example: Set A: {(1, 2), (2, 4), (3, 6), (4, 8)}.

Exit Ticket

Give students a graph of a relation. Ask them to: 1. State whether it is a function and explain how they know (referencing the vertical line test). 2. List the domain and range of the function using inequality notation.

Discussion Prompt

Pose the following to small groups: 'Imagine you are designing a system to recommend movies based on a user's viewing history. Can this recommendation system be a function? Why or why not? What would the domain and range be?'

Frequently Asked Questions

What is the difference between a relation and a function?

A relation is any set of ordered pairs connecting inputs to outputs. A function is a specific type of relation where each input is paired with exactly one output. A relation becomes a function when no input maps to more than one output. All functions are relations, but not all relations are functions.

What are domain and range in a function?

The domain is the complete set of all valid inputs for a function, and the range is the complete set of all resulting outputs. On a graph, the domain corresponds to the set of all x-values the graph covers, and the range corresponds to all y-values. Identifying domain and range from a graph requires reading the extent of the graph horizontally and vertically.

How does the vertical line test work?

Drag an imaginary vertical line across a graph from left to right. If the vertical line intersects the graph at more than one point at any x-value, the relation is not a function, because one input is paired with multiple outputs. If every vertical line touches the graph at most once, the graph represents a function.

How does active learning help students understand what a function is?

Students internalize the function definition best when they must construct their own examples and non-examples, not just classify given ones. Designing a mapping diagram that is a function, then deliberately breaking it, requires students to engage directly with the 'one input, one output' rule. This generative process leads to a more flexible understanding than classification drills alone.