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Function Notation: The Meaning of f(x)

Ninth graders learn function notation — the powerful f(x) symbol — understanding that it names a function and shows the input, and learning to evaluate f(x) by substituting a value for x.

Grade 9Functions55 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
Start the Lesson
Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Read function notation.
  • ✓Understand input and output.
  • ✓Evaluate f(x).
  • ✓Interpret f(x) in context.
Essential Question

You’ll see f(x) everywhere in algebra. It looks mysterious — but it just names a function and shows its input. What does f(x) really mean, and how do you use it?

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Lesson Phases
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Vocabulary Terms
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Standards Aligned
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Interactive Task
Model · Interactive Tool

The Function Machine

Project this and change the function and the input x. Watch f(x) evaluate — substitute the input, get the output!

⚙️ Evaluate f(x) — input in, output outTry it
a: 2
b: 3
input x: 4

The Lesson · Gradual Release (I Do · We Do · You Do)

55 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — The Mystery of f(x)

5 min

f(x) is everywhere in algebra — what does it actually mean?

👩‍🏫 Teacher Moves

  • Show f(x) notation.
  • Ask what it means.
  • Set the goal: function notation.

🎒 Student Actions

  • See f(x).
  • Wonder what it means.
  • Predict the idea.
2

I Do — Input and Output

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • f(x) names a function.
  • x is the input.
  • Evaluate by substituting.

🎒 Student Actions

  • Learn the notation.
  • See the input.
  • Substitute.
3

We Do — Explore Together

13 min

Class uses the Function Machine.

👩‍🏫 Teacher Moves

  • Change the function and input.
  • Evaluate f(x).
  • Read the output.

🎒 Student Actions

  • Set it.
  • Evaluate it.
  • Read output.
4

You Do — On Your Own

12 min

Students evaluate.

👩‍🏫 Teacher Moves

  • Evaluate f(x) for given inputs.
  • Interpret f(x) in context.
  • Find inputs from outputs.

🎒 Student Actions

  • Evaluate them.
  • Interpret them.
  • Work backward.
5

Close — Close

5 min

One evaluation.

👩‍🏫 Teacher Moves

  • Give a function and input.
  • Evaluate f(x).
  • Hand out the exit ticket.

🎒 Student Actions

  • Evaluate it.
  • Explain it.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School Algebra).

CCSS
F-IF.2

Use function notation and evaluate functions for inputs in their domains.

CCSS
F-IF.1

Understand a function as a rule assigning each input one output.

CCSS
F-IF.5

Relate the domain of a function to its graph.

CCSS
F-IF.4

Interpret key features of functions.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Function-machine visual.
  • Sentence frame: “f(___) means substitute ___ for x.”
  • Use an input-output table.

Support & Access

IEP / 504
  • Start with simple functions.
  • Substitute step by step.
  • Use a function machine.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Evaluate at an expression, f(a+1).
  • Work backward from output.
  • Interpret f(x) in a real context.
Materials

What to gather

  • 📽️Projector / board
  • 🧮Calculators
  • 💻The Function Machine
  • 📄Input-output tables
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

functiona rule assigning each input one outputfunction notationthe f(x) symbolinputthe value put into a function (x)outputthe result, f(x)evaluateto find the output for an inputsubstituteto put a value in for a variabledomainall allowed inputsrangeall possible outputs
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
If f(x) = 2x + 1, what is f(3)?
7 (2×3 + 1).
QUESTION 2
In f(x), what does the x represent?
The input to the function.
QUESTION 3
What does it mean to “evaluate” f(4)?
Substitute 4 in for x and compute the output.

Going deeper? Work backward.

Have students work in reverse — given f(x) = 3x − 2 and an output of 10, find the input x. A printable function-notation sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 Function NotationFlip
Card 1
Term
Tap to flip →
Meaning
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Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 Function NotationQuiz
Score: 0
1 / 6
Question 1
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Nice work!

Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ Function NotationPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: domain, evaluate, function, function notation, input, output, range, substitute
  1. to find the output for an input
  2. all allowed inputs
  3. the result, f(x)
  4. a rule assigning each input one output
  5. all possible outputs
  6. the value put into a function (x)
  7. to put a value in for a variable
  8. the f(x) symbol

Part B · Show what you learned

  1. If f(x) = 2x + 1, what is f(3)?
  2. In f(x), what does the x represent?
  3. What does it mean to “evaluate” f(4)?
Answer key — Part A: 1) evaluate · 2) domain · 3) output · 4) function · 5) range · 6) input · 7) substitute · 8) function notation
Part B: 1) 7 (2×3 + 1). 2) The input to the function. 3) Substitute 4 in for x and compute the output.