What is a function in math?

A function is a rule that assigns exactly one output to each input. If you put the same value in, you always get the same value out. The vertical line test checks this graphically: if any vertical line crosses the graph more than once, the relation is not a function.

What is the difference between domain and range?

The domain is the set of all valid inputs a function can accept. The range is the set of all outputs the function actually produces. For example, f(x) = x squared has domain all real numbers but range only zero and above, since squaring never produces a negative result.

What does one-to-one mean for a function?

A one-to-one function never repeats an output value -- distinct inputs always produce distinct outputs. This property is required for an inverse function to exist. Graphically, a function is one-to-one if every horizontal line crosses its graph at most once.

What are even and odd functions?

An even function satisfies f(-x) = f(x) and is symmetric about the y-axis. Examples include x squared and cosine. An odd function satisfies f(-x) = -f(x) and has rotational symmetry about the origin. Examples include x cubed and sine.

What are function transformations?

Transformations modify a parent function's graph by shifting it (translation), flipping it (reflection), or stretching and compressing it (dilation). The general form g(x) = a times f(b(x - h)) + k captures all four types, where h and k control position and a and b control shape.

Functions Terms and Definitions

About This Glossary

This glossary organizes 28 essential functions terms into six categories that build from foundational ideas to graphing and transformations.

Core Concepts establishes the foundation with 6 entries: function, relation, domain, range, independent variable, and dependent variable. These terms define what a function is, what goes in, and what comes out.

Types & Classification covers 6 entries on how functions are categorized: one-to-one, even, odd, increasing, decreasing, and piecewise functions. Each classification describes a structural property that determines behavior and eligibility for operations like inversion.

Operations & Inverses addresses 2 entries on combining and reversing functions: composition chains two functions in sequence, and the inverse function undoes what the original does.

Function Families presents 5 parent functions and their key properties: linear, quadratic, constant, absolute value, and the general concept of a parent function. Each family defines a characteristic shape that transformations modify.

Graph Features defines 4 entries on what you read from a graph: zeros (x-intercepts), asymptotes, local maxima, and local minima. These features describe the curve's intersections, boundaries, and turning points.

Transformations completes the glossary with 5 entries on modifying graphs: the general transformation framework, translations (shifts), reflections (flips), and dilations (stretches and compressions).

Each definition includes intuitive explanations, key properties, common errors, and links to detailed lesson pages. Use the search bar or category filters above to navigate.

Core ConceptsFunction FamiliesGraph FeaturesOperations & InversesTransformationsTypes & Classification

AAbsolute Value FunctionFunction Families AAsymptoteGraph Features CComposition of FunctionsOperations & Inverses CConstant FunctionFunction Families DDecreasing FunctionTypes & Classification DDependent VariableCore Concepts DDilationTransformations DDomainCore Concepts EEven FunctionTypes & Classification FFunctionCore Concepts IIncreasing FunctionTypes & Classification IIndependent VariableCore Concepts IInverse FunctionOperations & Inverses LLinear FunctionFunction Families LLocal MaximumGraph Features LLocal MinimumGraph Features OOdd FunctionTypes & Classification OOne-to-One FunctionTypes & Classification PParent FunctionFunction Families PPiecewise FunctionTypes & Classification QQuadratic FunctionFunction Families RRangeCore Concepts RReflectionTransformations RRelationCore Concepts TTransformationTransformations TTranslationTransformations ZZero of a FunctionGraph Features

Core Concepts(6)

Function Families(5)

Graph Features(4)

Operations & Inverses(2)

Transformations(4)

Types & Classification(6)

27 of 27 terms

Core Concepts

Function Relation Domain Range Independent Variable Dependent Variable

Function Families

Parent Function Linear Function Quadratic Function Constant Function Absolute Value Function

Graph Features

Zero of a Function Asymptote Local Maximum Local Minimum

Operations & Inverses

Composition of Functions Inverse Function

Transformations

Transformation Translation Reflection Dilation

Types & Classification

One-to-One Function Even Function Odd Function Increasing Function Decreasing Function Piecewise Function

27 terms

Core Concepts

(6 items)

Function

A rule that assigns exactly one output to each input. Formally, a function

f

from set

A

to set

B

is a mapping

f: A o B

such that every element of

A

is associated with precisely one element of

B

What is a Function

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intuitionnotationcommon errorsrelated concepts

A machine with one slot in and one slot out. Drop in a number, get back exactly one number — never two, never none. The same input always produces the same output.

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Relation

Any set of ordered pairs — a collection of input-output associations with no restriction on how many outputs an input may have.

Functions vs Relations

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intuitionexamplesrelated concepts

A relation is the broader category: any pairing of inputs with outputs, no rules about uniqueness. Every function is a relation, but a relation that assigns two outputs to one input is not a function.

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Domain

The set of all inputs for which a function produces a valid output:

\text{Dom}(f) = \{x \mid f(x) \text{ is defined}\}

What is Domain

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intuitionnotationspecial casesrelated concepts

The complete collection of values you are allowed to feed into the function. Anything that causes division by zero, an even root of a negative number, or a logarithm of a non-positive number is excluded.

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Range

The set of all output values a function actually produces as the input varies across the entire domain:

\text{Ran}(f) = \{f(x) \mid x \in ext{Dom}(f)\}

What is Range

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intuitionnotationcommon errorsrelated concepts

Everything the function can actually spit out. The domain is what goes in; the range is what comes out. Not every real number needs to appear —

f(x) = x^2

never outputs a negative number.

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Independent Variable

The input variable of a function, whose value is chosen freely from the domain.

Input, Output, and Variables

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intuitionnotationrelated concepts

The quantity you control. You pick its value; the function responds with an output. In

y = f(x)

, the variable

x

is independent — you decide what goes in.

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Dependent Variable

The output variable of a function, whose value is determined by the input through the function rule.

Input, Output, and Variables

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intuitionnotationrelated concepts

The quantity that responds. Its value depends entirely on what was fed in. In

y = f(x)

, the variable

y

is dependent — it has no choice; it is whatever

f

makes it.

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Types & Classification

(6 items)

One-to-One Function

A function where distinct inputs always produce distinct outputs:

f(a) = f(b) implies a = b

One-to-One Functions

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intuitionpropertiescommon errorsrelated concepts

No sharing allowed among outputs. Every output traces back to exactly one input, so the process can be reversed. This is the requirement for an inverse function to exist.

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Even Function

A function satisfying

f(-x) = f(x)

for all

x

in its domain.

Even and Odd Functions

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intuitionexamplespropertiesrelated concepts

Mirror symmetry about the

y

-axis. The left half of the graph is a perfect reflection of the right half. Replacing

x

with

-x

changes nothing.

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Odd Function

A function satisfying

f(-x) = -f(x)

for all

x

in its domain.

Even and Odd Functions

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intuitionexamplespropertiesrelated concepts

Rotational symmetry about the origin. Rotating the graph

180°

around

(0,0)

produces the same curve. Replacing

x

with

-x

negates the output.

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Increasing Function

A function is increasing on an interval if

a < b \implies f(a) < f(b)

for all

a, b

in that interval.

Increasing and Decreasing

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intuitionpropertiesrelated concepts

The graph rises from left to right. Larger inputs yield larger outputs — the function moves only upward as you scan rightward across the interval.

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Decreasing Function

A function is decreasing on an interval if

a < b \implies f(a) > f(b)

for all

a, b

in that interval.

Increasing and Decreasing

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intuitionpropertiesrelated concepts

The graph falls from left to right. Larger inputs yield smaller outputs — the function moves only downward as you scan rightward across the interval.

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Piecewise Function

A function defined by multiple formulas, each applying to a different subset of the domain.

What is a Piecewise Function

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intuitionnotationexamplesrelated concepts

Different rules for different regions. The function checks where the input falls and applies the corresponding formula. Each input still gets exactly one output — only the rule producing it depends on location.

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Operations & Inverses

(2 items)

Composition of Functions

The composition of

f

and

g

is the function

(f \circ g)(x) = f(g(x))

, where the output of

g

becomes the input of

f

What is Function Composition

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intuitionnotationpropertiesrelated concepts

Chaining two machines in series. The first machine processes the input; its output feeds directly into the second machine. The final result depends on both machines working in sequence.

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Inverse Function

A function

f^{-1}

that reverses

f

: if

f(a) = b

, then

f^{-1}(b) = a

. Equivalently,

f(f^{-1}(x)) = x

and

f^{-1}(f(x)) = x

What is an Inverse Function

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intuitionnotationpropertiesrelated concepts

The undo button. Whatever

f

does,

f^{-1}

reverses it. Doubling has halving as its inverse. Adding five has subtracting five. The round trip always returns to the starting point.

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Function Families

(5 items)

Parent Function

The simplest function in a family — the baseline shape with no transformations applied.

Parent Functions

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intuitionexamplesrelated concepts

The default template. Every other function in the family is a shifted, stretched, or reflected version of the parent. Knowing the parent means knowing the basic shape; transformations adjust position and scale.

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Linear Function

A function of the form

f(x) = mx + b

, where

m

is the slope and

b

is the

y

-intercept.

Linear Functions

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intuitionpropertiesrelated concepts

A straight line. The output changes at a constant rate — every unit increase in input produces exactly

m

units of change in output. No curvature, no surprises.

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Quadratic Function

A function of the form

f(x) = ax^2 + bx + c

with $a
eq 0$. The graph is a parabola.

Quadratic Functions

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intuitionpropertiesrelated concepts

A U-shaped curve (or inverted U). The function changes direction at the vertex — falling then rising, or rising then falling. The leading coefficient

a

controls which way the parabola opens and how wide it is.

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Constant Function

A function of the form

f(x) = c

where

c

is a fixed real number. Every input produces the same output.

Constant Functions

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intuitionpropertiesrelated concepts

A flat line. No matter what goes in, the same value comes out. The function ignores its input entirely.

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Absolute Value Function

The function

f(x) = |x|

, returning the distance of

x

from zero:

|x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases}

Absolute Value Function

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intuitionpropertiescommon errorsrelated concepts

Strips the sign. Positive inputs pass through unchanged; negative inputs become positive. The output is always the magnitude — how far from zero, regardless of direction.

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Graph Features

(4 items)

Zero of a Function

An input value

c

where

f(c) = 0

. Also called a root of the function.

Zeros and Roots

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intuitionpropertiesexamplesrelated concepts

Where the function's output hits zero — graphically, where the curve crosses or touches the

x

-axis. Finding zeros means solving

f(x) = 0

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Asymptote

A line that the graph of a function approaches but does not reach (or reaches only in the limit).

Asymptotes

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intuitionspecial casescommon errorsrelated concepts

A boundary the curve chases forever without arriving. The graph gets arbitrarily close to the line but never settles on it (or crosses it only finitely many times).

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Local Maximum

A function value

f(c)

such that

f(c) \geq f(x)

for all

x

in some open interval containing

c

Local Extrema

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intuitionpropertiesrelated concepts

A peak — the highest point in the immediate neighborhood. The curve rises to this point and then falls. Nearby outputs are all lower, though outputs farther away may be higher.

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Local Minimum

A function value

f(c)

such that

f(c) leq f(x)

for all

x

in some open interval containing

c

Local Extrema

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intuitionpropertiesrelated concepts

A valley — the lowest point in the immediate neighborhood. The curve falls to this point and then rises. Nearby outputs are all higher, though outputs farther away may be lower.

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Transformations

(4 items)

Transformation

An operation that modifies the graph of a function by shifting, reflecting, stretching, or compressing it, producing a new function from an existing one.

Transformations Overview

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intuitionnotationrelated concepts

Reshaping a curve without rebuilding it from scratch. Start with a known parent function and adjust its position, orientation, and scale. The underlying shape is preserved — only where and how it appears changes.

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Translation

A transformation that shifts the graph of a function horizontally, vertically, or both, without changing its shape, size, or orientation.

Vertical Translations

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intuitionpropertiescommon errorsrelated concepts

Sliding the entire curve to a new position. Every point moves the same distance in the same direction. The shape is unchanged — only the location shifts.

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Reflection

A transformation that flips the graph of a function across an axis, reversing its orientation in one direction.

Vertical Reflections

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intuitionpropertiesrelated concepts

A mirror image. Reflecting across the

x

-axis flips the graph upside down; reflecting across the

y

-axis swaps left and right.

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Dilation

A transformation that stretches or compresses the graph of a function, changing its scale without shifting or reflecting it.

Vertical Stretch and Compression

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intuitionpropertiescommon errorsrelated concepts

Pulling the curve away from an axis (stretch) or pushing it toward an axis (compression). The shape narrows or widens, steepens or flattens, but the basic form remains recognizable.

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