Algebra · StudentHub Lesson

Introduction to Functions

Understand functions as rules that map inputs to unique outputs, using function notation.

20 minIntermediate

What you will learn

Define a function as an input-output relationship
Use function notation like f(x)
Evaluate a function for a given input
Determine whether a relation is a function
Interpret function graphs

Watch the lesson

What is a function? | Functions and their graphs | Algebra II · Khan Academy

Watch on YouTube

Topic notes

Main Idea

A function is a rule that assigns exactly one output to each input. Function notation f(x) names the rule and shows what value is being evaluated.

Key Concepts

Each input (x-value) must map to exactly one output (y-value)
The vertical line test checks if a graph represents a function
Function notation: f(x) means 'the output of function f when the input is x'

Definitions

Function: a relation where each input has exactly one output
Domain: the set of all possible input values
Range: the set of all possible output values

Formulas

To evaluate f(a), substitute a wherever x appears in f(x)
Example: if f(x) = 2x+3, then f(4) = 2(4)+3 = 11

Examples

f(x) = x², find f(3): f(3) = 9
g(x) = 8x+2, find g(0): g(0)=2

Common Mistakes

Reading f(x) as f times x
Assuming any equation is a function without checking repeated inputs
Substituting incorrectly into multi-term expressions

Key concepts

Input-output relationshipFunction notation f(x)Domain and rangeVertical line testEvaluating functions

Important terms

Function: A rule assigning exactly one output to each input.
Domain: The set of allowed input values.
Range: The set of possible output values.

Worked examples

Problem

If f(x) = 3x - 1, find f(5)

1. Substitute x=5
2. f(5)=3(5)-1
3. f(5)=15-1

Answer: 14

Problem

If g(x) = x² + 2, find g(-2)

1. Substitute x=-2
2. g(-2)=(-2)²+2
3. g(-2)=4+2

Answer: 6

Quick revision

Each input maps to one output
f(x) notation names the rule
Domain = inputs, range = outputs
Vertical line test checks graphs
Substitute carefully when evaluating
A function can repeat outputs, not inputs

Check your understanding

Question 1 · Multiple choice

If f(x) = 2x + 1, what is f(3)?

Question 2 · Multiple choice

Which best defines a function?

Question 3 · Short answer

If h(x) = x² - 3, find h(4).

Question 4 · Short answer

What test determines if a graph is a function?

Question 5 · True or false

A function can have two different outputs for the same input.

Done with Introduction to Functions?

Mean, Median and Mode Arithmetic Sequences