Functions & Function Notation — I can determine whether a...

SECTION 1

Historical Context & Motivation

Long before anyone used the word "function," mathematicians were exploring relationships between quantities. Ancient Babylonian astronomers tracked how the length of daylight changed with the seasons, and Greek geometers studied how the area of a circle depends on its radius. These thinkers were already working with functions — they just didn't have a formal name or notation for them yet.

The concept of a function gradually crystallized over several centuries as mathematicians needed a precise way to describe how one quantity determines another. The notation we use today, like f(x), didn't appear overnight — it evolved through the work of some of history's greatest mathematical minds.

1637

Descartes Links Algebra and Geometry

René Descartes publishes La Géométrie, introducing the coordinate plane and showing that algebraic equations can describe geometric curves — an early step toward formalizing relationships between variables.

1694

Leibniz Coins the Word "Function"

Gottfried Wilhelm Leibniz first uses the Latin word functio to describe quantities that depend on a variable, giving mathematicians a shared vocabulary.

1734

Euler Introduces f(x) Notation

Leonhard Euler popularizes the notation f(x) to represent the output of a function for a given input x. This compact notation is still the standard in classrooms worldwide nearly 300 years later.

1837

Dirichlet's Modern Definition

Peter Gustav Lejeune Dirichlet refines the definition of a function: any rule that assigns exactly one output to each input. This broad definition opened the door to modern mathematics.

So why does this matter to you? Every time you calculate a tip at a restaurant, convert temperatures, or figure out how long a road trip will take at a certain speed, you're using a function. The question this lesson addresses is: How do we determine whether a relationship qualifies as a function, and how do we use function notation to communicate about it clearly?

SECTION 2

Core Principles & Definitions

Before we dive into function notation, we need to build a solid foundation with a few key ideas. A relation is any set of ordered pairs — it simply links inputs to outputs. A function is a special kind of relation that follows one strict rule: every input must produce exactly one output. That's the golden rule of functions.

Relation

A set of ordered pairs (x, y) that pairs inputs with outputs. A relation can pair one input with multiple outputs. Example: {(2, 3), (2, 5), (4, 7)}.

Function

A special relation where each input (x-value) maps to exactly one output (y-value). No input is allowed to have two different outputs.

Domain

The set of all possible input values (x-values) for a function. Think of it as all the values you're allowed to plug in.

Range

The set of all possible output values (y-values) that the function can produce. It's the collection of results you can get out.

Function Notation: f(x)

Read as "f of x." The letter f names the function, and x is the input variable. f(x) represents the output. For example, f(3) means "the output when 3 is the input."

✦ KEY TAKEAWAY

Think of a function like a vending machine. You press one button (input), and you get exactly one item (output). If you press button B4 and sometimes get chips and sometimes get a candy bar, that vending machine is broken — it's not a function! A working vending machine always gives you the same, single output for each input.

It's worth noting that a function can have two different inputs that produce the same output. For example, both 3 and −3 can produce 9 when squared. That's perfectly fine. The rule only goes one direction: each input gets one output, but different inputs can share an output.

SECTION 3

Visual Explanation — Mapping Diagrams

One of the best ways to see whether a relation is a function is through a mapping diagram. A mapping diagram shows the domain on the left, the range on the right, and arrows connecting each input to its output. If any single input has arrows pointing to two or more outputs, the relation is not a function.

The left mapping diagram shows a function: each input (1, 2, 3) maps to exactly one output. Note that inputs 2 and 3 both map to 5, which is allowed. The right diagram is not a function because input 1 maps to both 4 and 5.

When you look at graphs on a coordinate plane, you can use the Vertical Line Test. Imagine dragging a vertical line across the graph from left to right. If the vertical line ever crosses the graph in more than one point at the same time, the graph does not represent a function. This is because a vertical line at a particular x-value checks whether that input has multiple outputs.

SECTION 4

Mathematical Framework — Function Notation

You've probably been writing equations like y = 2x + 3 for a while. Function notation replaces y with a symbol like f(x), which gives us extra information. It tells us the name of the function (f), what the input variable is (x), and allows us to easily talk about specific outputs by substituting values.

FUNCTION NOTATION

f(x) = 2x + 3

f = the name of the function • x = the input variable • f(x) = the output (same as y)

The key advantage of function notation is that you can evaluate the function at specific inputs by replacing x with a number. When we write f(5), we mean "plug 5 in for x and calculate the result."

EVALUATING A FUNCTION

f(5) = 2(5) + 3 = 10 + 3 = 13

We substitute 5 for every x in the rule, then simplify. The result tells us the ordered pair (5, 13) is on the graph of f.

DIFFERENT FUNCTION NAMES

g(t) = t² − 4t + 1

Functions can use any letter for the name (f, g, h, etc.) and any variable for the input (x, t, n, etc.). The notation g(t) means "the function g with input t."

⚠️ Common Mistake

Students sometimes read f(x) as "f times x." It does not mean multiplication! The parentheses in function notation indicate that x is the input to the function f. Think of the parentheses as a container holding the input value.

SECTION 5

Multiple Representations of Functions

Functions can appear in many different forms: as equations, tables, graphs, mapping diagrams, or even verbal descriptions. No matter the representation, the same rule applies — each input must produce exactly one output. Let's look at how to check whether a relation is a function across several representations.

Methods for identifying functions across different representations

Representation	How to Check for a Function	Example
Set of Ordered Pairs	Look at the x-values (first coordinates). If any x-value appears more than once with different y-values, it's not a function.	{(1,3), (2,5), (3,3)} ✓ Function
Table of Values	Check the input column. If any input value repeats with a different output, it's not a function.	x: 1, 2, 2 → y: 4, 5, 6 ✗ Not a function
Graph	Apply the Vertical Line Test. If any vertical line touches the graph more than once, it's not a function.	A circle fails; a parabola opening up passes.
Equation	Solve for y in terms of x. If you can get two or more y-values for the same x, it's not a function.	y = x² ✓ \| x² + y² = 25 ✗
Mapping Diagram	Check that no element in the domain has more than one arrow leaving it.	See Section 3 diagram above.

The Vertical Line Test in action. On the left, a curve where every vertical line (dashed yellow) touches the graph at most once — this is a function. On the right, a circle where vertical lines cross the graph at two points — this is not a function.

SECTION 6

Worked Example

Let's work through a complete example that combines identifying a function and using function notation to evaluate it.

Given f(x) = 3x² − 2x + 1, determine whether this is a function and find f(−2).

Step 1 — Confirm It's a FunctionThe rule f(x) = 3x² − 2x + 1 is an equation where every input x produces exactly one output, because for any real number you substitute for x, the arithmetic gives a single result. Since x is squared, added, and combined with constants — all operations that produce one answer — this is a function.

✓ This is a function.

Step 2 — Substitute the InputTo find f(−2), replace every x in the expression with −2. Be careful with negative signs and exponents: f(−2) = 3(−2)² − 2(−2) + 1

f(−2) = 3(−2)² − 2(−2) + 1

Step 3 — Apply the Exponent FirstFollowing the order of operations (PEMDAS), handle the exponent first: (−2)² = 4. Note that the entire −2 is squared, so the result is positive. f(−2) = 3(4) − 2(−2) + 1

(−2)² = 4

Step 4 — MultiplyNow handle the multiplications: 3 × 4 = 12, and −2 × (−2) = 4 (a negative times a negative is positive). f(−2) = 12 + 4 + 1

3(4) = 12 and −2(−2) = +4

Step 5 — Add to Get the Final AnswerCombine the terms: 12 + 4 + 1 = 17. This means when the input is −2, the output is 17. In ordered-pair form, the point (−2, 17) lies on the graph of f.

f(−2) = 17

SECTION 7

Functions vs. Non-Functions — Comparisons

Understanding what makes something a function is easier when you compare functions with non-functions side by side. The table below highlights the key differences and common examples of each.

Side-by-side comparison of functions and non-functions

Feature	Function	Non-Function (Relation)
Rule	Each input → exactly one output	An input can map to multiple outputs
Vertical Line Test	Passes (every vertical line hits at most once)	Fails (some vertical line hits more than once)
Example Equations	y = 2x + 1, y = x², y = \|x\|	x² + y² = 9, x = y², y² = x + 4
Graph Shapes	Lines (non-vertical), parabolas opening up/down, V-shapes	Circles, ellipses, parabolas opening left/right
Function Notation	Can use f(x) notation because output is unique	Cannot use f(x) notation because output is ambiguous

✦ KEY TAKEAWAY

Think of a function like a smartphone contact list. Each person's name (input) is linked to exactly one phone number (output). If you had two different numbers stored under the same name with no way to tell them apart, your phone couldn't know which one to call — that's a non-function. Function notation like f(x) is like telling your phone, "Call the number stored under this specific contact name."

SECTION 8

Connection to Advanced Topics

The concepts you've learned here — identifying functions and using function notation — are the foundation for nearly everything you'll study in algebra and beyond. As you progress through math, these ideas expand in exciting ways.

How today's concepts connect to future coursework

This Lesson	Where It Leads
Evaluating f(x) at a single value	Analyzing f(x) across intervals, finding domain restrictions, and graphing entire function families
Identifying linear and quadratic functions	Studying exponential, logarithmic, trigonometric, and piecewise functions in Algebra 2 and Pre-Calculus
Using f(x) notation	Composing functions f(g(x)), finding inverse functions f⁻¹(x), and performing function transformations
Vertical Line Test on graphs	Horizontal Line Test to determine if a function has an inverse, one-to-one function analysis

In Algebra 2, you'll encounter function composition — plugging one function into another — and inverse functions, which "undo" what a function does. Both of these depend heavily on your comfort with function notation. The better you understand f(x) now, the smoother those future topics will feel.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

A student says that the relation {(3, 7), (5, 7), (8, 7)} is NOT a function because the output 7 repeats. Is the student correct? Explain your reasoning.

PROBLEM 2 — BASIC CALCULATION

Given g(x) = 4x − 7, find g(3) and g(−1).

PROBLEM 3 — INTERMEDIATE

Determine whether the following set of ordered pairs represents a function: {(−2, 4), (0, 1), (3, 9), (−2, 6), (5, 2)}. If it is not a function, identify which input causes the problem.

PROBLEM 4 — APPLIED

A rideshare app charges a base fare plus a per-mile rate. The total cost is modeled by C(m) = 2.50 + 1.75m, where m is the number of miles driven. Find the cost of a 12-mile ride, and explain what C(0) represents in this context.

PROBLEM 5 — CRITICAL THINKING

Consider the equation x² + y² = 25. Explain why this relation is not a function. Then describe how you could restrict it to create a function, and write the restricted version using function notation.

SUMMARY

Lesson Summary

A function is a special type of relation in which every input from the domain produces exactly one output in the range. You can test whether a relation is a function using mapping diagrams (no input has two arrows leaving it), tables and ordered pairs (no x-value repeats with a different y-value), or graphs (the Vertical Line Test).

Function notation like f(x) names the function (f), identifies the input (x), and represents the output. To evaluate a function at a specific value, substitute that value for every occurrence of the variable in the expression and simplify. Remember: f(x) does not mean f times x! These foundational skills — identifying functions and using function notation — are the building blocks for every function-related topic you'll encounter in future math courses.