Teacher Guide: Functions

Learning Objective 5-10 min lesson

Students will: Determine if a relation is a function, use function notation, evaluate functions for given inputs, and distinguish between linear and non-linear functions.

Why this matters for FAST: Functions are foundational for high school algebra. FAST tests require students to identify functions from tables and graphs, use function notation, and classify functions as linear or non-linear.

Materials Needed

Whiteboard/projector
Student Concept Worksheet
Graph paper
Function machine visual
Colored markers
Vertical line test cards

Common Misconceptions to Address

Misconception #1: All Relations Are Functions

Students think any set of ordered pairs is a function, not understanding that each input can only have ONE output.

How to Address:

"A function is like a vending machine - you press ONE button (input) and get ONE item (output). If pressing the same button sometimes gave you chips and sometimes gave you candy, that machine would be broken! In a function, each input has EXACTLY ONE output."

Misconception #2: Confusing f(x) with Multiplication

Students interpret f(x) as "f times x" instead of "f of x" or "the function f evaluated at x."

How to Address:

"f(x) is NOT multiplication! The parentheses mean 'of' not 'times.' Think of f(x) as 'the output when x goes into function f.' If f(x) = 2x + 1, then f(3) means 'plug in 3 for x': f(3) = 2(3) + 1 = 7."

Misconception #3: Confusing the Vertical Line Test

Students don't understand why the vertical line test works or apply it incorrectly.

How to Address:

"A vertical line represents ONE input (x-value). If the vertical line hits the graph more than once, that means one input has multiple outputs - which breaks the function rule! If EVERY vertical line only hits once, it's a function."

Lesson Steps

1

Define a Function (2 min)

SAY THIS:

"A FUNCTION is a special relationship where each INPUT has exactly ONE OUTPUT. Think of it as a machine: you put something in, and you always get the same thing out for that input."

Function Rule: Each x-value (input) can only have ONE y-value (output)!

Function: {(1, 3), (2, 5), (3, 7)} - Each x has one y ✓

NOT a Function: {(1, 3), (1, 5), (2, 7)} - x = 1 has TWO y's ✗

2

Identifying Functions from Tables (2 min)

IS a Function

x	1	2	3
y	4	7	10

Each x has one y

NOT a Function

x	1	2	1
y	4	7	9

x = 1 has TWO outputs!

3

Function Notation (2 min)

SAY THIS:

"Instead of y = 2x + 1, we can write f(x) = 2x + 1. They mean the same thing! f(x) is read as 'f of x' and means 'the output when we input x.' To find f(3), substitute 3 for x."

Example: If f(x) = 3x - 2, find f(5).

f(5) = 3(5) - 2 = 15 - 2 = 13

The input is 5, the output is 13.

4

Vertical Line Test for Graphs (2 min)

Vertical Line Test: If ANY vertical line crosses the graph more than once, it's NOT a function.

Function

Every vertical line hits once

Not a Function (circle)

Vertical line hits twice

5

Linear vs Non-Linear (2 min)

Linear functions: Graph is a straight line, equation has no exponents on variables (y = mx + b).

Non-linear functions: Graph is curved, equation has exponents (y = x², y = 2^x) or other non-linear forms.

Check for Understanding

Quick Exit Ticket:

"If f(x) = 4x - 3, what is f(2)?"

Solution: f(2) = 4(2) - 3 = 8 - 3 = 5

IXL Skills to Assign

Recommended IXL Practice:

Identify functions Find outputs from graphs Evaluate functions Complete function tables Linear or nonlinear functions