Function Machine Handout
Algebra 1 Guided Notes Unit: Functions & Relations
Function Notation & The Function Machine
Demystifying inputs, outputs, and how mathematicians name their rules
Name:
Date: Period:
PART 1
Cracking the Notation: What Does \(f(x)\) Mean?
Function notation replaces "\(y =\)"
Anatomy of \(f(x)\)
\(f\) ( \(x\) ) = Output
\(f\) = Name of function rule • \(x\) = Input fed inside (Domain)
\(f(x)\) = Resulting output value (Range • replaces \(y\))
How to Read It Out Loud
Read "\(f(x)\)" as: "f of x"
Critical Warning: Parentheses do NOT mean multiplication! \(f(x)\) never means "\(f\) times \(x\)".
Real-Life Functions: Letters Have Meaning! In the real world, mathematicians pick letters that tell a story:
h(a) A Child's Height is a function of Age
Input a = age (years) • Output h(a) = height (inches)
h(10) = 54 means: "At age 10, the child is 54 inches tall."
p(h) Your Pay is a function of the Hours you worked
Input h = hours worked • Output p(h) = total pay ($)
p(15) = 225 means: "Working 15 hours earns a paycheck of $225."
PART 2
The Function Machine in Action: 3 Inputs → 3 Outputs
Trace each colored input through the machine rule
3 Inputs into "IN": 5, 8, 20
Machine Placard Rule: "Triple input and add 6"
Algebraic Function Rule: \(f(x) = 3x + 6\)
\(f(5)\) = \(3(5) + 6\) = 15 + 6 =
Pair: (5, )
\(f(8)\) = \(3(8) + 6\) = =
Pair: (8, )
\(f(20)\) = \(3(20) + 6\) = =
Pair: (20, )
The Golden Rule: Every single input fed into "IN" produces exactly one output exiting "OUT"!
PART 3
Testing Zero & Negative Inputs: Machine \(f(x) = 3x + 6\)
| Input Value \((x)\) |
Step-by-Step Substitution |
Output \(f(x)\) |
Ordered Pair \((x, y)\) |
| \(f(0)\) |
\(3(0) + 6 = 0 + 6\) |
\(6\) |
\((0, 6)\) |
| \(f(-2)\) |
\(3(-2) + 6 = -6 + 6\) |
\(0\) |
\((-2, 0)\) |
| \(f(-4)\) |
\(3(-4) + 6 = -12 + 6\) |
\(-6\) |
\((-4, -6)\) |
Algebra 1 • Handout: Function Machine & Notation Page 1 of 2 • Flip over for Multi-Operation Practice & TI-Nspire™ Lab →
Algebra 1 Guided Notes
Part 2: Multiple Operations & The TI-Nspire Handheld
Student Name:
PART 4
Independent Run: Linear Machine \(g(x) = 2x - 5\)
Substitute carefully and simplify
Problem 1: Find \(g(4)\)
Show substitution & steps:
\(g(4) =\) Pair:
Problem 2: Find \(g(-3)\)
Show substitution & steps:
\(g(-3) =\) Pair:
Problem 3: Find \(g(0)\)
Show substitution & steps:
\(g(0) =\) Pair:
PART 5
Exponents & Multi-Operations: Machine \(h(x) = x^2 + 1\)
Parentheses Shield: Wrap negative inputs in parentheses when squaring!
Notice: \((-4)^2 = (-4)(-4) = +16\), whereas \(-4^2 = -(4 \cdot 4) = -16\).
Problem 4: Find \(h(3)\)
Show substitution & steps:
\(h(3) =\)
Problem 5: Find \(h(-4)\)
Use parentheses shield!
\(h(-4) =\)
Problem 6: Find \(h(0)\)
Show substitution & steps:
\(h(0) =\)
PART 6
Technology Bridge: Defining Functions on TI-Nspire™ CX
1
Open Calculator Page: Press home → select 1: New Document → Add Calculator.
2
Define the Function: Type:
Define f(x) = 3x + 6
Shortcut: Press menu → 1: Actions → 1: Define. Handheld says "Done".
3
Evaluate Instantly: Type f(20) and press enter! Handheld computes 66!
*Doc 1.1 • Calculator RAD AUTO REAL
Define f(x) = 3·x + 6
Done
f(5)
21
f(20)
66
PART 7
Hands-On Handheld Lab: Verify & Conquer
Work by hand first, then check on TI-Nspire
Define these two rules on your calculator: Define k(x) = 4x - 7 | Define p(x) = x² - 3x
Task A: Evaluate \(k(-3)\) Linear
By Hand Substitution:
Hand Output: Nspire:
Task B: Evaluate \(p(5)\) Quadratic
By Hand Substitution:
Hand Output: Nspire:
Think & Reflect: Why use function notation instead of just "\(y =\)"?
When dealing with multiple equations like \(k(x)\) and \(p(x)\), why is function notation clearer than having multiple equations both called "\(y = \dots\)"?
Algebra 1 • Handout: Function Machine & Notation Page 2 of 2 • Mastery Check Complete
Function Machine Answer Key
Teacher Answer Key & Facilitation Guide Algebra 1 Guided Notes 3.1
Function Notation & The Function Machine
Complete solutions, exemplar explanations, and instructional tips
Red Text = Student Answers
Handout Page 1 Solutions
PART 1 SOLUTIONS
Cracking the Notation: What Does \(f(x)\) Mean?
Key Takeaways & Pronunciation
Anatomy Breakdown
\(f\) ( \(x\) ) = Output (\(y\))
\(f\) = Name of function rule • \(x\) = Input fed inside (Domain)
\(f(x)\) = Resulting output value (Range • completely replaces \(y\))
Teacher Talk & Misconception Alert
Have the entire class choral repeat: "f of x"! Warn students: "Do not distribute \(f\)! Parentheses here mean 'input container', not multiplication."
Prompt: "If \(f(3) = 11\), what goes into the machine? (3) What comes out? (11)"
Real-Life Functions: Letters Have Meaning! Exemplar contextual translations:
h(a) A Child's Height is a function of Age
Input a = age (years) • Output h(a) = height (inches)
h(10) = 54 means: "At age 10, the child is 54 inches tall."
p(h) Your Pay is a function of the Hours you worked
Input h = hours worked • Output p(h) = total pay ($)
p(15) = 225 means: "Working 15 hours earns a paycheck of $225."
PART 2 SOLUTIONS
The Function Machine in Action: 3 Inputs → 3 Outputs
Rule: \(f(x) = 3x + 6\)
3 Inputs into "IN": 5, 8, 20
Machine Placard Rule: "Triple input and add 6"
Algebraic Rule: \(f(x) = 3x + 6\)
\(f(5)\) = \(3(5) + 6\) = 15 + 6 = 21
Pair: (5, 21)
\(f(8)\) = \(3(8) + 6\) = 24 + 6 = 30
Pair: (8, 30)
\(f(20)\) = \(3(20) + 6\) = 60 + 6 = 66
Pair: (20, 66)
Golden Rule Debrief: Emphasize that ordered pairs \((5, 21), (8, 30), (20, 66)\) are points ready to be plotted on the Cartesian plane!
PART 3 SOLUTIONS
Testing Zero & Negative Inputs: Machine \(f(x) = 3x + 6\)
| Input Value \((x)\) |
Step-by-Step Substitution |
Output \(f(x)\) |
Ordered Pair \((x, y)\) |
| \(f(0)\) |
\(3(0) + 6 = 0 + 6\) |
6 |
(0, 6) |
| \(f(-2)\) |
\(3(-2) + 6 = -6 + 6\) |
0 |
(-2, 0) |
| \(f(-4)\) |
\(3(-4) + 6 = -12 + 6\) |
-6 |
(-4, -6) |
Algebra 1 • Teacher Answer Key: Function Machine & Notation Page 1 of 2 • Flip over for Multi-Operation & TI-Nspire™ Solutions →
Teacher Answer Key
Part 2: Multiple Operations & The TI-Nspire Handheld
Handout Page 2 Solutions
PART 4 SOLUTIONS
Independent Run: Linear Machine \(g(x) = 2x - 5\)
Work Shown & Ordered Pairs
Problem 1: Find \(g(4)\)
Exemplar student work:
\(g(4) = 2(4) - 5\)
\(= 8 - 5 = 3\)
\(g(4) =\) 3 Pair: (4, 3)
Problem 2: Find \(g(-3)\)
Exemplar student work:
\(g(-3) = 2(-3) - 5\)
\(= -6 - 5 = -11\)
\(g(-3) =\) -11 Pair: (-3, -11)
Problem 3: Find \(g(0)\)
Exemplar student work:
\(g(0) = 2(0) - 5\)
\(= 0 - 5 = -5\)
\(g(0) =\) -5 Pair: (0, -5)
PART 5 SOLUTIONS
Quadratic Machine: \(h(x) = x^2 + 1\)
Common Error on Problem 5: Students often type -4^2 + 1 into calculators and get -15. Explain that the handheld follows PEMDAS: \(4^2 = 16\), then applies negative \(\implies -16\). Wrapping as (-4)^2 squares the negative number!
Problem 4: Find \(h(3)\)
Work:
\(h(3) = (3)^2 + 1\)
\(= 9 + 1 = 10\)
\(h(3) =\) 10
Problem 5: Find \(h(-4)\)
Work (Shield used!):
\(h(-4) = (-4)^2 + 1\)
\(= 16 + 1 = 17\)
\(h(-4) =\) 17
Problem 6: Find \(h(0)\)
Work:
\(h(0) = (0)^2 + 1\)
\(= 0 + 1 = 1\)
\(h(0) =\) 1
PART 6 GUIDE
Technology Bridge: TI-Nspire™ CX Facilitation Notes
Handheld Syntax Tips for the Teacher:
• Implicit Multiplication: TI-Nspire handles 3x fine, but with multiple letters like ab, it assumes a single variable name. Remind students to use the multiplication key 3 · x if they get unexpected outputs.
• Shortcut Key: Press ctrl + := to define functions directly: f(x):=3x+6.
*Doc 1.1 • Calculator RAD AUTO REAL
Define f(x) = 3·x + 6
Done
f(5)
21
f(20)
66
PART 7 SOLUTIONS
Hands-On Handheld Lab: Complete Verification
Expected student outputs
Target handheld rules: Define k(x) = 4x - 7 | Define p(x) = x² - 3x
Task A: Evaluate \(k(-3)\) Linear
\(k(-3) = 4(-3) - 7 = -12 - 7 = -19\)
Hand Output: -19 Nspire: -19
Task B: Evaluate \(p(5)\) Quadratic
\(p(5) = (5)^2 - 3(5) = 25 - 15 = 10\)
Hand Output: 10 Nspire: 10
Exemplar Response for Think & Reflect:
Why is function notation clearer than multiple equations both called "\(y = \dots\)"?
Model Student Response: "Function notation gives each machine its own unique name (like \(k\) or \(p\)). If both equations were written as \(y = \dots\), you wouldn't know which equation to substitute numbers into! On the TI-Nspire, typing \(k(2)\) tells the calculator exactly which formula to compute without having to rewrite it."
Algebra 1 • Teacher Answer Key: Function Machine & Notation Page 2 of 2 • Handout Key Complete
