Algebra Formulas

Function Arithmetic

Composition

Inverse Functions

Symmetry

Transformations

Linear Function Forms

Quadratic Function Forms

Asymptotes

Rates of Change

38 formulas

Function Arithmetic

(6 formulas)

Sum of Functions

(f + g)(x) = f(x) + g(x)

Sum of Functions

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Adds two functions pointwise: at each input, evaluate both functions and add the outputs. Addition is commutative and associative, mirroring number arithmetic.

Difference of Functions

(f - g)(x) = f(x) - g(x)

Difference of Functions

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Subtracts

g

from

f

pointwise. Order matters:

(f - g)(x) = -(g - f)(x)

. Subtraction is neither commutative nor associative.

Product of Functions

(fg)(x) = f(x) \cdot g(x)

Product of Functions

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Multiplies two functions pointwise: at each input, evaluate both and multiply the outputs. Multiplication is commutative, associative, and distributes over addition.

Quotient of Functions

\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}

Quotient of Functions

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Divides

f

g

pointwise. The denominator function

g

must be nonzero at the input — values of

x

where

g(x) = 0

are excluded from the domain even if both

f

and

g

are defined there.

Domain of a Combined Function

\text{Dom}(f + g) = \text{Dom}(f - g) = \text{Dom}(fg) = \text{Dom}(f) \cap \text{Dom}(g)

Domain of Combined Functions

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For sums, differences, and products, the domain is the intersection of the component domains. An input must be valid for both functions in order for the combined operation to produce an output.

Domain of a Quotient

\text{Dom}\!\left(\frac{f}{g}\right) = \bigl(\text{Dom}(f) \cap \text{Dom}(g)\bigr) \setminus \{x : g(x) = 0\}

Additional Restrictions for Quotient

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The quotient inherits all restrictions from both functions plus one more: any input where

g

equals zero is removed. Even if algebraic simplification appears to eliminate the restriction, the original domain exclusion remains — a hole, not a valid point.

Composition

(4 formulas)

Composition of Functions

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

What is Function Composition

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Chains two functions in sequence:

g

acts first on the input, then

f

acts on

g

's output. Read inside-out:

g

is the inner function,

f

is the outer function. Composition is generally not commutative —

f \circ g

and

g \circ f

are different functions.

Domain of a Composition

\text{Dom}(f \circ g) = \{x \in \text{Dom}(g) : g(x) \in \text{Dom}(f)\}

Domain of Composite Functions

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An input

x

is valid for

f \circ g

only if two conditions both hold:

x

must be in the domain of the inner function

g

, and the output

g(x)

must be a valid input for the outer function

f

. Either failure removes

x

from the composite domain.

Composition Associativity

(f \circ g) \circ h = f \circ (g \circ h)

Composing More Than Two Functions

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Composition is associative: when chaining three or more functions, grouping does not affect the result. Both sides equal

f(g(h(x)))

. Order still matters — associativity does not imply commutativity.

Inverse of a Composition

(f \circ g)^{-1} = g^{-1} \circ f^{-1}

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The inverse of a composition reverses the order of the factors. To undo "first

g

, then

f

," apply the inverses in the opposite order: first undo

f

, then undo

g

. Like reversing a sequence of nested operations.

Inverse Functions

(4 formulas)

Inverse Function Definition

f(a) = b \iff f^{-1}(b) = a

What is an Inverse Function

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The inverse function

f^{-1}

reverses the input-output pairing of

f

. Whatever

f

sends to

b

, the inverse sends back to its source. Exists only when

f

is one-to-one.

Inverse Composition Property

f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x

Verifying Inverses via Composition

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Composing a function with its inverse — in either order — yields the identity. This is the operational definition of "inverse" and the standard verification: if both compositions reduce to

x

, the two functions are inverses.

Inverse of an Inverse

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

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Inverting an inverse returns the original function. The reverse of the reverse is the forward direction.

Domain Range Swap of Inverse

\text{Dom}(f^{-1}) = \text{Ran}(f) \quad \text{and} \quad \text{Ran}(f^{-1}) = \text{Dom}(f)

Domain and Range Swap

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Inputs and outputs trade places. Every output of

f

becomes an input of

f^{-1}

, and every input of

f

becomes an output of

f^{-1}

. Graphically, this matches the reflection of the curve across the line

y = x

Symmetry

(2 formulas)

Even Function Test

f(-x) = f(x) \quad \text{for all } x \in \text{Dom}(f)

Even and Odd Functions

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An even function is unchanged when its input is negated. The graph is symmetric about the

y

-axis: the left half mirrors the right.

Odd Function Test

f(-x) = -f(x) \quad \text{for all } x \in \text{Dom}(f)

Even and Odd Functions

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An odd function negates its output when its input is negated. The graph has rotational symmetry about the origin: rotating 180° about the origin produces the same curve. If

0

is in the domain, then

f(0) = 0

Transformations

(7 formulas)

General Transformation

g(x) = a \cdot f(b(x - h)) + k

Transformations Overview

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The master transformation form. Four parameters control how the parent graph is reshaped:

h

shifts horizontally,

k

shifts vertically,

a

scales/reflects vertically,

b

scales/reflects horizontally. Inside-the-function operations affect

x

; outside-the-function operations affect

y

Vertical Translation

g(x) = f(x) + k

Vertical Translations

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Shifts the graph up by

k

units (or down if

k

is negative). The shape is preserved; only the vertical position changes. Affects range, not domain.

Horizontal Translation

g(x) = f(x - h)

Horizontal Translations

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Shifts the graph right by

h

units (or left if

h

is negative). The direction is counterintuitive: subtracting inside the argument moves the graph in the positive direction. Affects domain, not range.

Vertical Reflection

g(x) = -f(x)

Vertical Reflections

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Flips the graph across the

x

-axis. Each output is negated: peaks become valleys, valleys become peaks.

x

-intercepts are fixed; range is negated.

Horizontal Reflection

g(x) = f(-x)

Horizontal Reflections

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Flips the graph across the

y

-axis. Each input is negated before the function acts.

y

-intercept is fixed; domain is negated. An even function is unchanged by this reflection.

Vertical Dilation

g(x) = a \cdot f(x)

Vertical Stretch and Compression

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Scales the graph vertically.

|a| > 1

: stretches taller.

0 < |a| < 1

: compresses shorter.

a < 0

: also reflects across the

x

-axis. The

x

-intercepts are fixed; the

y

-intercept and range scale by

a

Horizontal Dilation

g(x) = f(bx)

Horizontal Stretch and Compression

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Scales the graph horizontally.

|b| > 1

: compresses narrower (counterintuitive).

0 < |b| < 1

: stretches wider.

b < 0

: also reflects across the

y

-axis. The

y

-intercept is fixed; domain scales by

1/b

Linear Function Forms

(5 formulas)

Slope Intercept Form

f(x) = mx + b

Linear Functions

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The most common form of a linear function. The coefficient

m

is the slope (rate of change); the constant

b

is the

y

-intercept (output when input is zero).

Point Slope Form

y - y_1 = m(x - x_1)

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Builds the equation of a line from a known slope

m

and a known point

(x_1, y_1)

on the line. Convenient when slope and a single point are given directly.

Standard Form Linear

Ax + By = C

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An implicit form of a line, with

x

and

y

on the same side. Useful for systems of linear equations and for representing vertical lines, which slope-intercept form cannot capture.

Slope Formula

m = \frac{y_2 - y_1}{x_2 - x_1}

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Computes the slope of a line through two points

(x_1, y_1)

and

(x_2, y_2)

. Rise divided by run — the change in output per unit change in input.

Perpendicular Slopes

m_1 \cdot m_2 = -1

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Two non-vertical lines are perpendicular if and only if the product of their slopes equals

-1

. Equivalently, each slope is the negative reciprocal of the other.

Quadratic Function Forms

(5 formulas)

Standard Form Quadratic

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

Quadratic Functions

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The general polynomial form of a quadratic function. The coefficient

a

controls opening direction (upward if

a > 0

, downward if

a < 0

) and width. The constant

c

is the

y

-intercept.

Vertex Form Quadratic

f(x) = a(x - h)^2 + k

Quadratic Functions

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Places the vertex of the parabola explicitly at

(h, k)

. Reading transformations directly from this form: horizontal shift by

h

, vertical shift by

k

, vertical scale by

a

. Convenient for graphing and for finding extrema.

Vertex from Coefficients

h = -\frac{b}{2a}, \quad k = f(h)

Quadratic Functions

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Gives the vertex coordinates of a parabola from the standard form

ax^2 + bx + c

. The

x

-coordinate is

-b/(2a)

; the

y

-coordinate is the function evaluated at that

x

Axis of Symmetry

x = -\frac{b}{2a}

Quadratic Functions

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The vertical line passing through the vertex, about which the parabola is mirror-symmetric. Same

x

-coordinate as the vertex.

Factored Form Quadratic

f(x) = a(x - r_1)(x - r_2)

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Expresses a quadratic using its roots

r_1

and

r_2

directly. Reveals the

x

-intercepts at a glance. Available only when the quadratic has real roots.

Asymptotes

(3 formulas)

Vertical Asymptote Rational

x = a \quad \text{where } Q(a) = 0 \text{ and } P(a) \neq 0

Rational Functions

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For a rational function

\frac{P(x)}{Q(x)}

, vertical asymptotes occur at zeros of the denominator that are not also zeros of the numerator. At these points, the function blows up to

\pm\infty

. Common factors that cancel produce holes, not asymptotes.

Horizontal Asymptote Equal Degree

y = \frac{a_n}{b_m} \quad \text{when } \deg P = \deg Q

Rational Functions

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When the numerator and denominator have the same degree, the rational function approaches the ratio of the leading coefficients as

x \to \pm\infty

. The horizontal asymptote sits at this ratio.

Horizontal Asymptote Numerator Lower

y = 0 \quad \text{when } \deg P < \deg Q

Rational Functions

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When the numerator has lower degree than the denominator, the rational function approaches zero as

x \to \pm\infty

. The

x

-axis itself is the horizontal asymptote.

Rates of Change

(2 formulas)

Average Rate of Change

\frac{f(b) - f(a)}{b - a}

Average Rate of Change

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Measures how much the output changes per unit of input over an interval

[a, b]

. Geometrically, the slope of the secant line through

(a, f(a))

and

(b, f(b))

Difference Quotient

\frac{f(x + h) - f(x)}{h}

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The average rate of change of

f

over an interval of width

h

starting at

x

. Shrinking

h

toward zero produces the instantaneous rate of change — the foundation of the derivative in calculus.

Function Arithmetic

Sum of Functions Difference of Functions Product of Functions Quotient of Functions Domain of a Combined Function Domain of a Quotient

Composition

Composition of Functions Domain of a Composition Composition Associativity Inverse of a Composition

Inverse Functions

Inverse Function Definition Inverse Composition Property Inverse of an Inverse Domain Range Swap of Inverse

Symmetry

Even Function Test Odd Function Test

Transformations

General Transformation Vertical Translation Horizontal Translation Vertical Reflection Horizontal Reflection Vertical Dilation Horizontal Dilation

Linear Function Forms

Slope Intercept Form Point Slope Form Standard Form Linear Slope Formula Perpendicular Slopes

Quadratic Function Forms

Standard Form Quadratic Vertex Form Quadratic Vertex from Coefficients Axis of Symmetry Factored Form Quadratic

Asymptotes

Vertical Asymptote Rational Horizontal Asymptote Equal Degree Horizontal Asymptote Numerator Lower

Rates of Change

Average Rate of Change Difference Quotient