Algebra Mastery
The Human Knowledge Project
Chapter 9 — Functions
9.1 Learning Objectives
By the end of this chapter, you should be able to:
understand what functions represent
distinguish functions from general relationships
identify inputs and outputs
evaluate functions correctly
interpret function notation
determine whether a relation is a function
understand domain and range intuitively
recognize linear and nonlinear functions
connect functions to real-world systems
develop structural understanding of mathematical mappings
9.2 Big Picture — Functions Describe Input and Output Systems
Functions are one of the most important ideas in all of mathematics.
A function describes a system where:
an input enters
a rule operates
an output emerges
Examples:
typing on a calculator
entering values into a computer program
measuring temperature over time
calculating wages
converting units
predicting motion
Functions allow mathematics to model:
change
dependency
transformation
causality
computation
Modern science, engineering, computing, and artificial intelligence rely heavily on functions.
9.3 What Is a Function?
A function is a relationship where:
each input has exactly one output
Example:
f(x) = 2x + 1
This means:
take x
multiply by 2
add 1
Example:
If:
x = 3
then:
f(3) = 2(3) + 1 = 7
9.4 Inputs and Outputs
Functions connect:
inputs
outputs
Example:
Input Output
1 3
2 5
3 7
4 9
Rule:
f(x) = 2x + 1
Functions are essentially:
mathematical machines
9.5 Function Notation
Function notation helps describe operations clearly.
Example:
f(x)
means:
the output of function f using input x
Example:
f(x) = x² + 2
Evaluate:
f(3)
Substitute:
3² + 2 = 11
Result:
f(3) = 11
9.6 Why Function Notation Matters
Function notation allows mathematics to:
organize systems
describe transformations
model computation
build higher mathematics
Functions become the foundation for:
calculus
physics
machine learning
engineering
programming
9.7 Evaluating Functions
Example:
f(x) = 3x - 4
Find:
f(5)
Substitute:
3(5) - 4
Simplify:
15 - 4 = 11
9.8 Multiple Inputs
Functions can accept many possible inputs.
Example:
f(x) = x + 2
x f(x)
1 3
2 4
5 7
-3 -1
Functions define entire patterns of behavior.
9.9 Functions vs Non-Functions
A relation is NOT a function if:
one input produces multiple outputs
Function Example
(1,3)
(2,5)
(3,7)
Each input has ONE output.
This IS a function.
Non-Function Example
(2,4)
(2,9)
Input:
2
produces TWO outputs.
This is NOT a function.
9.10 The Vertical Line Test
Graphs help identify functions.
Rule:
if a vertical line crosses a graph more than once,
the graph is NOT a function
Why?
Because:
one x-value
would produce multiple y-values
9.11 Domain
The domain is:
the set of allowable inputs
Example:
f(x) = 1/x
x cannot equal:
0
because division by zero is undefined.
So:
0 is excluded from the domain
9.12 Range
The range is:
the set of possible outputs
Example:
f(x) = x²
Possible outputs:
0
1
4
9
16
Notice:
outputs are never negative
9.13 Linear Functions
Linear functions have:
constant rate of change
straight-line graphs
Example:
f(x) = 2x + 1
Linear functions are predictable and extremely important.
9.14 Nonlinear Functions
Nonlinear functions do NOT produce straight lines.
Example:
f(x) = x²
Graph:
curves upward
Nonlinear systems appear constantly in nature.
9.15 Real-World Functions
Examples:
Situation Function
wages pay(hours)
temperature T(time)
population P(year)
distance d(time)
memory usage M(processes)
Functions model dependency relationships.
9.16 Functions and Programming
Computer programs rely heavily on functions.
Example:
input → processing → output
This is fundamentally functional behavior.
Functions connect algebra to computing directly.
9.17 Common Beginner Difficulties
Students often struggle with:
substitution errors
function notation confusion
input/output reversal
evaluating expressions incorrectly
domain misunderstandings
These struggles are normal.
Function intuition develops through:
repetition
experimentation
visualization
pattern recognition
9.18 Mental Model
A function is like:
a machine
a process
a transformation system
Input enters.
Rules operate.
Output emerges.
9.19 Warm-Up Problems
Problems
Evaluate:
f(x) = x + 2
f(3)
Evaluate:
f(x) = 2x
f(5)
Evaluate:
f(x) = x²
f(4)
Evaluate:
f(x) = 3x - 1
f(2)
Evaluate:
f(x) = x² + 2
f(3)
Evaluate:
f(x) = 4x + 5
f(0)
Evaluate:
f(x) = x - 7
f(10)
Evaluate:
f(x) = 2x²
f(2)
Determine whether relation is a function:
(1,2)
(2,3)
(3,4)
Determine whether relation is a function:
(2,5)
(2,7)
Identify domain issue:
f(x) = 1/x
Identify whether function is:
linear
nonlinear
f(x) = 5x + 1
9.20 Guided Problems
Problems
Evaluate:
f(x) = 2x + 3
f(7)
Evaluate:
f(x) = x² - 4
f(5)
Evaluate:
f(x) = 3x + 1
f(-2)
Evaluate:
f(x) = x² + x
f(4)
Determine whether function is:
linear
nonlinear
f(x) = x²
Determine whether relation is a function:
(1,4)
(2,5)
(3,6)
Determine whether relation is a function:
(4,1)
(4,9)
Identify domain restriction:
f(x) = 1/(x - 2)
Explain function notation.
Explain the difference between:
input
output
Describe the vertical line test.
Explain why functions are important.
9.21 Challenge Problems
Evaluate:
f(x) = 2x² - 3x + 1
f(4)
Evaluate:
f(x) = x³
f(-2)
Evaluate:
f(x) = 5 - 2x
f(6)
Evaluate:
f(x) = x² - 2x + 1
f(3)
Determine whether graph would pass the vertical line test:
circle
Determine whether graph would pass the vertical line test:
straight line
Explain why some relations are not functions.
Explain the meaning of domain and range.
Describe a real-world system modeled by functions.
Explain why functions are fundamental in science, engineering, and computing.
9.22 Solutions
Solutions to Warm-Up Problems
5
10
16
5
11
5
3
8
function
not a function
x cannot equal 0
linear
Solutions to Guided Problems
17
21
-5
20
nonlinear
function
not a function
x cannot equal 2
Function notation describes the output of a function for a given input.
Inputs enter the function.
Outputs emerge after the rule operates.
If a vertical line crosses a graph more than once, the graph is not a function.
Functions model relationships, transformations, and systems throughout mathematics and science.
Solutions to Challenge Problems
21
-8
-7
4
A circle fails the vertical line test because some x-values produce multiple y-values.
A non-vertical straight line passes the vertical line test.
Some relations assign multiple outputs to the same input, violating the definition of a function.
Domain describes allowable inputs.
Range describes possible outputs.
Possible examples include:
wages
population growth
temperature changes
computer processing
fuel consumption
network performance
Functions provide structured models for change, dependency, computation, and prediction across science and technology.