Introduction to Functions: Input and Output

Functions come alive when students physically manipulate inputs and outputs, because abstract rules become concrete actions. Active learning lets students test their own assumptions about what defines a function, correcting misconceptions through immediate feedback rather than abstract explanations.

• JC 1
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Identify the domain, codomain, and range of a given function represented by an equation, table, or graph.
2. Determine if a relation is a function by applying the vertical line test to its graph.
3. Analyze whether a function is one-to-one by examining its domain and range, or by applying the horizontal line test.
4. Construct a composite function fg, specifying the domain of the composite function based on the domains of f and g.
5. Calculate the inverse of a function algebraically and verify its existence using the horizontal line test.

Think-Pair-Share

⏱ 25 min·Pairs

Card Sort: Function Matching

Prepare cards with inputs, outputs, and rules like f(x)=x^2. In pairs, students match valid pairs to form tables, discarding any input-output pair that violates the one-output rule. Pairs then identify domain restrictions and share one non-function example with the class.

Prepare & details

How do the concepts of domain, codomain, and range precisely define a function, and why does the choice of domain affect whether a function is one-to-one?

Facilitation Tip: For Card Sort: Function Matching, circulate to listen for student reasoning about rejected pairs, asking guiding questions like, 'Why did you remove this one? What rule did it break?'

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 30 min·Small Groups

Relay Race: Horizontal Line Test

Divide into small groups with a whiteboard. One student sketches a graph per turn, next adds a horizontal line to test for one-to-one. Groups race to classify five graphs correctly, discussing errors as a class.

Prepare & details

Under what conditions does a function possess an inverse, and how can you verify this both algebraically and graphically using the horizontal line test?

Facilitation Tip: For Relay Race: Horizontal Line Test, set a timer for each station so groups must move quickly, forcing them to use the horizontal line test intuitively rather than slowly calculating points.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 35 min·Whole Class

Function Machine: Inputs and Outputs

Select one student as the 'machine' with a secret rule. Class calls inputs; machine responds with outputs. Class guesses rule, plots points, and verifies domain-range via table. Rotate machines twice.

Prepare & details

Construct the composite function fg and determine the conditions on the domains of f and g that must hold for fg to be well-defined.

Facilitation Tip: For Function Machine: Inputs and Outputs, provide calculators only if students request them, encouraging mental math to reinforce the connection between rule and computation.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 40 min·Small Groups

Chain Build: Composite Functions

Small groups get cards for f and g. They chain outputs of f into g, noting domain conditions for fg to work. Groups present chains on posters, class verifies with sample inputs.

Prepare & details

How do the concepts of domain, codomain, and range precisely define a function, and why does the choice of domain affect whether a function is one-to-one?

Facilitation Tip: For Chain Build: Composite Functions, assign roles (input writer, operator, output checker) to ensure all students participate and catch errors before they compound.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Start with the function machine to build intuition: students see a rule applied to inputs and observe the outputs. Avoid rushing to formal definitions; let students discover the need for the vertical line test through their own mistakes. When teaching inverses, emphasize the horizontal line test as a tool for prediction, not just a rule to memorize. Research shows that students grasp domain restrictions better when they encounter real limits in equations before generalizing to all functions.

What to Expect

Success means students can distinguish valid functions from invalid relations, justify their choices using the vertical line test, and explain why domain choices matter for inverses. They should articulate the difference between codomain and range and confidently apply the horizontal line test to determine one-to-one mappings.

Watch Out for These Misconceptions

Common Misconception

During Card Sort: Function Matching, watch for students who pair inputs with multiple outputs without rejecting them.

What to Teach Instead

Circulate and ask groups to explain their pairing choices, prompting them to reexamine pairs where one input maps to two outputs. Encourage them to physically separate these pairs and discuss why they do not represent functions.

Common Misconception

During Function Machine: Inputs and Outputs, watch for students who assume the domain includes all real numbers unless told otherwise.

What to Teach Instead

Provide equations like f(x)=1/x and ask students to list valid inputs before computing outputs. Have them justify why certain inputs are excluded, linking domain to real-world feasibility.

Common Misconception

During Relay Race: Horizontal Line Test, watch for students who assume all functions have inverses.

What to Teach Instead

At the station with a non-one-to-one function, have students sketch the graph and physically draw horizontal lines to test. Ask them to explain why the lack of one-to-one mapping prevents an inverse.

Assessment Ideas

Quick Check

After Card Sort: Function Matching, give students three relations and ask them to sort them into 'Function' and 'Not a Function' piles. Collect their reasoning to assess understanding of the vertical line test.

Exit Ticket

During Function Machine: Inputs and Outputs, have students complete a half-sheet with f(x)=2x+1, domain {1, 2, 3}. Ask for the range, whether it is one-to-one, and a sentence explaining their reasoning based on the machine's outputs.

Discussion Prompt

After Chain Build: Composite Functions, present g(x)=x^2 and h(x)=x^3. Ask students to predict how domain choices affect one-to-one mapping and inverses, using their function machine and horizontal line test experiences to justify their answers.

Extensions & Scaffolding

• Challenge students to create their own function machines for peers, designing rules that include domain restrictions and verifying inverses with the horizontal line test.
• For students who struggle, provide partial function machine tables with some outputs missing, asking them to deduce the rule and complete the table.
• Deeper exploration: Have students research real-world functions (e.g., tax brackets, shipping costs) and present how domain choices affect their outputs and inverses.

Key Vocabulary

Domain	The set of all possible input values for which a function is defined.
Codomain	The set of all possible output values that a function could potentially produce.
Range	The set of all actual output values that a function produces for its given domain.
One-to-one function	A function where each output value corresponds to exactly one input value, meaning no two distinct inputs map to the same output.
Composite function	A function formed by applying one function to the output of another function, denoted as fg(x) = f(g(x)).