Functions Diagrams

One output per input

Functions Basics: Definition, Notation & Tests › What is a Function

A parabola passes, a circle fails

Functions Basics: Definition, Notation & Tests › The Vertical Line Test

f(x) = √x: domain [0, ∞) read off the axis

Analyzing Functions: Behavior & Intervals › Reading Domain and Range from Graph

Falling, then rising

Analyzing Functions: Behavior & Intervals › Intervals of Increase and Decrease

Local peaks against the highest point

Analyzing Functions: Behavior & Intervals › Locating Extrema from Graph

f(x) = x²: mirror symmetry about the y-axis

Analyzing Functions: Behavior & Intervals › Identifying Symmetry

Reading the period from two peaks

Analyzing Functions: Behavior & Intervals › Identifying Periodicity

f(x) = e^(−x): the curve flattens onto y = 0

Analyzing Functions: Behavior & Intervals › End Behavior from Graph

Adding outputs, one input at a time

Function Arithmetic: Operations on Functions › Sum of Functions

Where a product vanishes

Function Arithmetic: Operations on Functions › Product of Functions

A hole where the divisor is zero

Function Arithmetic: Operations on Functions › Quotient of Functions

The overlap of two domains

Function Arithmetic: Operations on Functions › Domain of Combined Functions

f(x) = x², g(x) = sin x: f(g(x)) = (sin x)²

Function Composition › What is Function Composition

Inside out, in both orders

Function Composition › Evaluating Compositions at a Point

f composed with itself

Function Composition › Composition with Itself

f(x) = eˆx, g(x) = ln x: f(g(x)) = x

Function Composition › Composition and Inverse Functions

Two graph readings, one composition

Function Composition › Graphical Composition

Nothing to rule out

Domain of a Function - How to Find It › What is Domain

What the formula allows, what the situation allows

Domain of a Function - How to Find It › Natural Domain vs Restricted Domain

f(x) = 1/x: every real except 0

Domain of a Function - How to Find It › Finding Domain: Rational Functions

f(x) = √x: inputs from 0 onward

Domain of a Function - How to Find It › Finding Domain: Radical Functions

f(x) = ln x: inputs strictly greater than 0

Domain of a Function - How to Find It › Finding Domain: Logarithmic Functions

Two restrictions, one domain

Domain of a Function - How to Find It › Finding Domain: Combined Functions

A straight line, constant slope

Function Families: Types, Graphs & Properties › Linear Functions

f(x) = 0.3x² − 3: a parabola

Function Families: Types, Graphs & Properties › Quadratic Functions

A cubic with a peak and a valley

Function Families: Types, Graphs & Properties › Cubic Functions

f(x) = 1/x: two branches and two asymptotes

Function Families: Types, Graphs & Properties › Rational Functions

Starts at the origin, grows ever more slowly

Function Families: Types, Graphs & Properties › Square Root Function

f(x) = |x|: a V with its corner at the origin

Function Families: Types, Graphs & Properties › Absolute Value Function

Constant pieces with a jump

Function Families: Types, Graphs & Properties › Step Functions

f(x) = 2ˆx: doubling with each unit step

Function Families: Types, Graphs & Properties › Exponential Functions

The logarithm's vertical wall

Function Families: Types, Graphs & Properties › Logarithmic Functions

f(x) = sin x: a wave of period 2π

Function Families: Types, Graphs & Properties › Trigonometric Functions

Four quadrants, one plotted point

Graphs of Functions › The Coordinate Plane

Five points of the graph of x squared

Graphs of Functions › The Graph of a Function

Two readings of the same graph

Graphs of Functions › Reading Function Values from Graphs

f(x) = 0.2x³ − 2x: the S-shape of a cubic

Graphs of Functions › Types of Curves

One function can be run backwards, the other cannot

Inverse Functions › What is an Inverse Function

f(x) = x³ and f⁻¹(x) = ∛x, mirrored in y = x

Inverse Functions › Inverse Function Notation

Coordinates swapped, points mirrored

Inverse Functions › Inverse as Reflection over y = x

f(x) = √x and f⁻¹(x) = x² for x ≥ 0

Inverse Functions › Domain and Range Swap

A round trip in both orders

Inverse Functions › Verifying Inverses via Composition

The parabola fails, the cubic passes

Inverse Functions › The Horizontal Line Test

f(x) = x² restricted to x ≥ 0: inverse √x

Inverse Functions › Restricting Domain to Create Inverse

f(x) = eˆx and f⁻¹(x) = ln x

Inverse Functions › Common Inverse Pairs

One function, two formulas

Piecewise Functions - Definition, Graphing & Examples › What is a Piecewise Function

Graph each piece on its own interval

Piecewise Functions - Definition, Graphing & Examples › Graphing Piecewise Functions

Two intervals, one domain with a gap

Piecewise Functions - Definition, Graphing & Examples › Domain of Piecewise Functions

Collecting the outputs of each piece

Piecewise Functions - Definition, Graphing & Examples › Range of Piecewise Functions

x + 1 for x < 0, x − 1 for x ≥ 0: a jump at 0

Piecewise Functions - Definition, Graphing & Examples › Continuity of Piecewise Functions

|x| as two pieces: −x then x

Piecewise Functions - Definition, Graphing & Examples › Absolute Value as Piecewise

The Heaviside step: −1 then 1

Piecewise Functions - Definition, Graphing & Examples › Step Functions

The plumber's charge as a graph

Piecewise Functions - Definition, Graphing & Examples › Writing Piecewise Functions

f(x) = x³: symmetric about the origin

Properties of Functions › Even and Odd Functions

f(x) = x²: symmetric about the y-axis

Properties of Functions › Symmetry

f(x) = x³: every horizontal line meets it once

Properties of Functions › One-to-One Functions

f(x) = eˆx: the outputs never reach 0 or below

Properties of Functions › Onto Functions

Falling, then rising

Properties of Functions › Increasing and Decreasing

eˆx and ln x: both only ever rise

Properties of Functions › Monotonic Functions

f(x) = sin x: every output between −1 and 1

Properties of Functions › Boundedness

A cubic’s peak and valley

Properties of Functions › Local Extrema

f(x) = 1/x: vertical x = 0 and horizontal y = 0

Properties of Functions › Asymptotes

The leading term decides the ends

Properties of Functions › End Behavior

f(x) = sin x repeats every 2π

Properties of Functions › Periodicity

Crossing and touching zeros

Properties of Functions › Zeros and Roots

Average rate of change as a slope

Properties of Functions › Average Rate of Change

Bending down, then bending up

Properties of Functions › Concavity

The domain shapes the range

Range of a Function › What is Range

f(x) = x³: the range is the whole codomain

Range of a Function › Range vs Codomain

f(x) = 1/x: every output except 0

Range of a Function › Finding Range Algebraically

f(x) = eˆx: approaching 0, never reaching it

Range of a Function › Finding Range by Analyzing Behavior

f(x) = x²: range [0, ∞) read off the y-axis

Range of a Function › Finding Range from Graph

f(x) = |x|: outputs from 0 upward

Range of a Function › Range of Common Function Types

g(x) = f(x) + 3: the parabola and its range move up by 3

Range of a Function › Effect of Transformations on Range

eˆx and ln x: range and domain trade places

Range of a Function › Range and Invertibility

All four parameters at once

Function Transformations: Shifts, Reflections & Stretches › Transformations Overview

g(x) = f(x) + 3

Function Transformations: Shifts, Reflections & Stretches › Transformation Notation

g(x) = f(x − 3)

Function Transformations: Shifts, Reflections & Stretches › Horizontal Translations

g(x) = −x²: reflected in the x-axis

Function Transformations: Shifts, Reflections & Stretches › Vertical Reflections

g(x) = √(−x): reflected in the y-axis

Function Transformations: Shifts, Reflections & Stretches › Horizontal Reflections

g(x) = 2f(x)

Function Transformations: Shifts, Reflections & Stretches › Vertical Stretch and Compression

g(x) = f(2x)

Function Transformations: Shifts, Reflections & Stretches › Horizontal Stretch and Compression

Four transformations, one parabola

Function Transformations: Shifts, Reflections & Stretches › Combining Transformations

Reading the equation off the graph

Function Transformations: Shifts, Reflections & Stretches › Writing Equations from Transformed Graphs

ln(x), frozen

Function Asymptotes Visualizer › The Logarithmic Family

tan(x), frozen

Function Asymptotes Visualizer › The Tangent Family

e⁻ˣ, frozen

Function Asymptotes Visualizer › Exponential Decay

1/(1 + x²), frozen

Function Asymptotes Visualizer › The Bell Curve

arctan(x), frozen

Function Asymptotes Visualizer › The Arctangent Function

1/(1 + e⁻ˣ), frozen

Function Asymptotes Visualizer › The Logistic Curve

1/x, frozen

Function Asymptotes Visualizer › The Reciprocal Function

(x + 1)/(x − 1), frozen

Function Asymptotes Visualizer › The Shifted Rational Function

x/(x² − 1), frozen

Function Asymptotes Visualizer › The Three-Branch Rational Function

x + 1/x, frozen

Function Asymptotes Visualizer › The Classic Oblique Function

(x² − 1)/x, frozen

Function Asymptotes Visualizer › The Oblique Rational Function

f(x) = x, domain frozen

Domain of a Function Visualizer › The Identity Function

f(x) = 2x, domain frozen

Domain of a Function Visualizer › The Scaled Linear Function

f(x) = x², domain frozen

Domain of a Function Visualizer › The Quadratic Function

f(x) = x³, domain frozen

Domain of a Function Visualizer › The Cubic Function

f(x) = eˣ, domain frozen

Domain of a Function Visualizer › The Exponential Function

f(x) = sin(x), domain frozen

Domain of a Function Visualizer › The Sine Function

f(x) = cos(x), domain frozen

Domain of a Function Visualizer › The Cosine Function

f(x) = |x|, domain frozen

Domain of a Function Visualizer › The Absolute Value Function

ln(x), domain frozen

Domain of a Function Visualizer › The Logarithm and Its Open Boundary

√x, domain frozen

Domain of a Function Visualizer › The Square Root and Its Closed Boundary

1/x, domain frozen

Domain of a Function Visualizer › The Reciprocal and Its Excluded Point

f(x) = x, range frozen

Range of a Function Visualizer › The Identity Function

f(x) = 2x, range frozen

Range of a Function Visualizer › The Scaled Linear Function

f(x) = x³, range frozen

Range of a Function Visualizer › The Cubic Function

f(x) = ln(x), range frozen

Range of a Function Visualizer › The Logarithmic Function

f(x) = x², range frozen

Range of a Function Visualizer › The Quadratic Function

f(x) = |x|, range frozen

Range of a Function Visualizer › The Absolute Value Function

f(x) = √x, range frozen

Range of a Function Visualizer › The Square Root Function

f(x) = eˣ, range frozen

Range of a Function Visualizer › The Exponential Function

f(x) = sin(x), range frozen

Range of a Function Visualizer › The Sine Function

f(x) = cos(x), range frozen

Range of a Function Visualizer › The Cosine Function

f(x) = 1/x, range frozen

Range of a Function Visualizer › The Reciprocal and Its Missing Output

x²: even, frozen

Function Symmetry Visualizer › Symmetry of the Quadratic

|x|: even, frozen

Function Symmetry Visualizer › Symmetry of the Absolute Value

cos(x): even, frozen

Function Symmetry Visualizer › Symmetry of Cosine

x: odd, frozen

Function Symmetry Visualizer › Symmetry of the Identity

x³: odd, frozen

Function Symmetry Visualizer › Symmetry of the Cubic

sin(x): odd, frozen

Function Symmetry Visualizer › Symmetry of Sine

1/x: odd, frozen

Function Symmetry Visualizer › Symmetry of the Reciprocal

√x: neither, frozen

Function Symmetry Visualizer › Symmetry of the Square Root

eˣ: neither, frozen

Function Symmetry Visualizer › Symmetry of the Exponential

ln(x): neither, frozen

Function Symmetry Visualizer › Symmetry of the Logarithm

x² + x: neither, frozen

Function Symmetry Visualizer › Symmetry of the Mixed Polynomial

y = x, tangent at x₀ = 2

Tangent Line Visualizer › Tangent to the Identity

y = 2x, tangent at x₀ = 1

Tangent Line Visualizer › Tangent to the Scaled Linear

x², tangent at x₀ = 1.5

Tangent Line Visualizer › Tangent to the Quadratic

x³, tangent at x₀ = 1

Tangent Line Visualizer › Tangent to the Cubic

1/x, tangent at x₀ = 1

Tangent Line Visualizer › Tangent to the Reciprocal

eˣ, tangent at x₀ = 1

Tangent Line Visualizer › Tangent to the Exponential

ln(x), tangent at x₀ = 2

Tangent Line Visualizer › Tangent to the Logarithm

√x, tangent at x₀ = 4

Tangent Line Visualizer › Tangent to the Square Root

|x|, tangent at x₀ = 2

Tangent Line Visualizer › Tangent to the Absolute Value

sin(x), tangent at x₀ = 0

Tangent Line Visualizer › Tangent to Sine

cos(x), tangent at x₀ = π/2

Tangent Line Visualizer › Tangent to Cosine

x-axis flip, frozen

Function Reflections Visualizer › Reflection Across the X-Axis

y-axis flip, frozen

Function Reflections Visualizer › Reflection Across the Y-Axis

Diagonal mirror, frozen

Function Reflections Visualizer › Reflection Across the Line y = x

Mirror at y = 1, frozen

Function Reflections Visualizer › Reflection Across a Horizontal Line

Mirror at x = 1, frozen

Function Reflections Visualizer › Reflection Across a Vertical Line

|f(x)|, frozen

Function Reflections Visualizer › Output Reflection: |f(x)|

f(|x|), frozen

Function Reflections Visualizer › Input Reflection: f(|x|)

a = 2, frozen

Transformations of Functions Visualizer › The Vertical Scale

k = 3, frozen

Transformations of Functions Visualizer › The Vertical Shift

b = 2, frozen

Transformations of Functions Visualizer › The Horizontal Scale

h = 3, frozen

Transformations of Functions Visualizer › The Horizontal Shift

All four at once, frozen

Transformations of Functions Visualizer › Combining All Four Transformations

f(x) = x, frozen

Functions Families Gallery › The Linear Family

f(x) = 0.3x² − 3, frozen

Functions Families Gallery › The Quadratic Family

f(x) = 0.2x³ − 2x, frozen

Functions Families Gallery › The Cubic Family

f(x) = x², frozen

Functions Families Gallery › The Power Family

f(x) = 1/x, frozen

Functions Families Gallery › The Rational Family

f(x) = 2ˣ, frozen

Functions Families Gallery › The Exponential Family

f(x) = ln(x), frozen

Functions Families Gallery › The Logarithmic Family

f(x) = sin(x), frozen

Functions Families Gallery › The Sine Family

f(x) = cos(x), frozen

Functions Families Gallery › The Cosine Family

f(x) = tan(x), frozen

Functions Families Gallery › The Tangent Family

f(x) = |x|, frozen

Functions Families Gallery › The Absolute Value Family

f(x) = √x, frozen

Functions Families Gallery › The Square Root Family

Identity ∘ quadratic, frozen

Function Composition Visualizer › Composing with the Identity

x² and sin(x), both orders

Function Composition Visualizer › Composing with the Quadratic

x³ and sin(x), both orders

Function Composition Visualizer › Composing with the Cubic

1/x and sin(x), both orders

Function Composition Visualizer › Composing with the Reciprocal

eˣ and sin(x), both orders

Function Composition Visualizer › Composing with the Exponential

ln(x) and x², both orders

Function Composition Visualizer › Composing with the Logarithm

sin(x) and x², both orders

Function Composition Visualizer › Composing with Sine

cos(x) and x², both orders

Function Composition Visualizer › Composing with Cosine

|x| and sin(x), both orders

Function Composition Visualizer › Composing with the Absolute Value

√x and sin(x), both orders

Function Composition Visualizer › Composing with the Square Root

eˣ ∘ ln(x) and ln ∘ eˣ, frozen

Function Composition Visualizer › The Inverse Pair: Exponential and Logarithm

√(x²) versus (√x)², frozen

Function Composition Visualizer › The Square and the Square Root

x² ∘ x², frozen

Function Composition Visualizer › Composing a Function with Itself

f(x) = x, frozen

Inverse Function Visualizer › The Identity and Its Inverse

2x and x/2, frozen

Inverse Function Visualizer › The Scaled Linear Function and Its Inverse

x³ and ∛x, frozen

Inverse Function Visualizer › The Cubic and the Cube Root

1/x, self-inverse, frozen

Inverse Function Visualizer › The Reciprocal and Its Inverse

eˣ and ln(x), frozen

Inverse Function Visualizer › The Exponential and the Logarithm

ln(x) and eˣ, frozen

Inverse Function Visualizer › The Logarithm and the Exponential

x² restricted, and √x, frozen

Inverse Function Visualizer › The Quadratic and the Square Root

√x and its inverse, frozen

Inverse Function Visualizer › The Square Root and Its Inverse

|x| restricted, frozen

Inverse Function Visualizer › The Absolute Value and Its Inverse