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Functions Formulas

38 formulas
Function ArithmeticGo to
CompositionGo to
Inverse FunctionsGo to
SymmetryGo to
TransformationsGo to
Linear Function FormsGo to
Quadratic Function FormsGo to
AsymptotesGo to
Rates of ChangeGo to

Function Arithmetic

(6 formulas)

Sum of Functions

(f+g)(x)=f(x)+g(x)(f + g)(x) = f(x) + g(x)
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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.
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Difference of Functions

(f−g)(x)=f(x)−g(x)(f - g)(x) = f(x) - g(x)
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Subtracts gg from ff pointwise. Order matters: (f−g)(x)=−(g−f)(x)(f - g)(x) = -(g - f)(x). Subtraction is neither commutative nor associative.
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Product of Functions

(fg)(x)=f(x)⋅g(x)(fg)(x) = f(x) \cdot g(x)
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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.
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Quotient of Functions

(fg)(x)=f(x)g(x)\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}
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Divides ff by gg pointwise. The denominator function gg must be nonzero at the input — values of xx where g(x)=0g(x) = 0 are excluded from the domain even if both ff and gg are defined there.
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Domain of a Combined Function

Dom(f+g)=Dom(f−g)=Dom(fg)=Dom(f)∩Dom(g)\text{Dom}(f + g) = \text{Dom}(f - g) = \text{Dom}(fg) = \text{Dom}(f) \cap \text{Dom}(g)
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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.
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Domain of a Quotient

Dom ⁣(fg)=(Dom(f)∩Dom(g))∖{x:g(x)=0}\text{Dom}\!\left(\frac{f}{g}\right) = \bigl(\text{Dom}(f) \cap \text{Dom}(g)\bigr) \setminus \{x : g(x) = 0\}
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The quotient inherits all restrictions from both functions plus one more: any input where gg equals zero is removed. Even if algebraic simplification appears to eliminate the restriction, the original domain exclusion remains — a hole, not a valid point.
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Composition

(4 formulas)

Composition of Functions

(f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))
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Chains two functions in sequence: gg acts first on the input, then ff acts on gg's output. Read inside-out: gg is the inner function, ff is the outer function. Composition is generally not commutative — f∘gf \circ g and g∘fg \circ f are different functions.
Function machine
xx
g(x)g(x)
f(g(x))f(g(x))
apply gg
apply ff
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Domain of a Composition

Dom(f∘g)={x∈Dom(g):g(x)∈Dom(f)}\text{Dom}(f \circ g) = \{x \in \text{Dom}(g) : g(x) \in \text{Dom}(f)\}
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An input xx is valid for f∘gf \circ g only if two conditions both hold: xx must be in the domain of the inner function gg, and the output g(x)g(x) must be a valid input for the outer function ff. Either failure removes xx from the composite domain.
Function machine
xx
g(x)g(x)
x∈Dom(f∘g)x \in \text{Dom}(f \circ g)
check x∈Dom(g)x \in \text{Dom}(g)
check g(x)∈Dom(f)g(x) \in \text{Dom}(f)
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Composition Associativity

(f∘g)∘h=f∘(g∘h)(f \circ g) \circ h = f \circ (g \circ h)
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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)))f(g(h(x))). Order still matters — associativity does not imply commutativity.
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Inverse of a Composition

(f∘g)−1=g−1∘f−1(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 gg, then ff," apply the inverses in the opposite order: first undo ff, then undo gg. Like reversing a sequence of nested operations.
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Inverse Functions

(4 formulas)

Inverse Function Definition

f(a)=b  ⟺  f−1(b)=af(a) = b \iff f^{-1}(b) = a
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The inverse function f−1f^{-1} reverses the input-output pairing of ff. Whatever ff sends to bb, the inverse sends back to its source. Exists only when ff is one-to-one.
Function machine
aa
bb
apply ff
apply f−1f^{-1}
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Inverse Composition Property

f(f−1(x))=xandf−1(f(x))=xf(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x
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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 xx, the two functions are inverses.
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Inverse of an Inverse

(f−1)−1=f(f^{-1})^{-1} = f
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Inverting an inverse returns the original function. The reverse of the reverse is the forward direction.
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Domain Range Swap of Inverse

Dom(f−1)=Ran(f)andRan(f−1)=Dom(f)\text{Dom}(f^{-1}) = \text{Ran}(f) \quad \text{and} \quad \text{Ran}(f^{-1}) = \text{Dom}(f)
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Inputs and outputs trade places. Every output of ff becomes an input of f−1f^{-1}, and every input of ff becomes an output of f−1f^{-1}. Graphically, this matches the reflection of the curve across the line y=xy = x.
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Symmetry

(2 formulas)

Even Function Test

f(−x)=f(x)for all x∈Dom(f)f(-x) = f(x) \quad \text{for all } x \in \text{Dom}(f)
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An even function is unchanged when its input is negated. The graph is symmetric about the yy-axis: the left half mirrors the right.
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Odd Function Test

f(−x)=−f(x)for all x∈Dom(f)f(-x) = -f(x) \quad \text{for all } x \in \text{Dom}(f)
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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 00 is in the domain, then f(0)=0f(0) = 0.
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Transformations

(7 formulas)

General Transformation

g(x)=a⋅f(b(x−h))+kg(x) = a \cdot f(b(x - h)) + k
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The master transformation form. Four parameters control how the parent graph is reshaped: hh shifts horizontally, kk shifts vertically, aa scales/reflects vertically, bb scales/reflects horizontally. Inside-the-function operations affect xx; outside-the-function operations affect yy.
Function machine
xx
x−hx - h
b(x−h)b(x - h)
f(b(x−h))f(b(x - h))
g(x)g(x)
shift: x−hx - h
scale: b(x−h)b(x - h)
apply ff
scale & shift: a ⋅+ka\,\cdot + k
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Vertical Translation

g(x)=f(x)+kg(x) = f(x) + k
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Shifts the graph up by kk units (or down if kk is negative). The shape is preserved; only the vertical position changes. Affects range, not domain.
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Horizontal Translation

g(x)=f(x−h)g(x) = f(x - h)
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Shifts the graph right by hh units (or left if hh is negative). The direction is counterintuitive: subtracting inside the argument moves the graph in the positive direction. Affects domain, not range.
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Vertical Reflection

g(x)=−f(x)g(x) = -f(x)
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Flips the graph across the xx-axis. Each output is negated: peaks become valleys, valleys become peaks. xx-intercepts are fixed; range is negated.
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Horizontal Reflection

g(x)=f(−x)g(x) = f(-x)
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Flips the graph across the yy-axis. Each input is negated before the function acts. yy-intercept is fixed; domain is negated. An even function is unchanged by this reflection.
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Vertical Dilation

g(x)=a⋅f(x)g(x) = a \cdot f(x)
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Scales the graph vertically. ∣a∣>1|a| > 1: stretches taller. 0<∣a∣<10 < |a| < 1: compresses shorter. a<0a < 0: also reflects across the xx-axis. The xx-intercepts are fixed; the yy-intercept and range scale by aa.
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Horizontal Dilation

g(x)=f(bx)g(x) = f(bx)
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Scales the graph horizontally. ∣b∣>1|b| > 1: compresses narrower (counterintuitive). 0<∣b∣<10 < |b| < 1: stretches wider. b<0b < 0: also reflects across the yy-axis. The yy-intercept is fixed; domain scales by 1/b1/b.
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Linear Function Forms

(5 formulas)

Slope Intercept Form

f(x)=mx+bf(x) = mx + b
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The most common form of a linear function. The coefficient mm is the slope (rate of change); the constant bb is the yy-intercept (output when input is zero).
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Point Slope Form

y−y1=m(x−x1)y - y_1 = m(x - x_1)
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Builds the equation of a line from a known slope mm and a known point (x1,y1)(x_1, y_1) on the line. Convenient when slope and a single point are given directly.
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Standard Form Linear

Ax+By=CAx + By = C
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An implicit form of a line, with xx and yy on the same side. Useful for systems of linear equations and for representing vertical lines, which slope-intercept form cannot capture.
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Slope Formula

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
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Computes the slope of a line through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2). Rise divided by run — the change in output per unit change in input.
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Perpendicular Slopes

m1⋅m2=−1m_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 −1-1. Equivalently, each slope is the negative reciprocal of the other.
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Quadratic Function Forms

(5 formulas)

Standard Form Quadratic

f(x)=ax2+bx+cf(x) = ax^2 + bx + c
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The general polynomial form of a quadratic function. The coefficient aa controls opening direction (upward if a>0a > 0, downward if a<0a < 0) and width. The constant cc is the yy-intercept.
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Vertex Form Quadratic

f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k
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Places the vertex of the parabola explicitly at (h,k)(h, k). Reading transformations directly from this form: horizontal shift by hh, vertical shift by kk, vertical scale by aa. Convenient for graphing and for finding extrema.
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Vertex from Coefficients

h=−b2a,k=f(h)h = -\frac{b}{2a}, \quad k = f(h)
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Gives the vertex coordinates of a parabola from the standard form ax2+bx+cax^2 + bx + c. The xx-coordinate is −b/(2a)-b/(2a); the yy-coordinate is the function evaluated at that xx.
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Axis of Symmetry

x=−b2ax = -\frac{b}{2a}
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The vertical line passing through the vertex, about which the parabola is mirror-symmetric. Same xx-coordinate as the vertex.
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Factored Form Quadratic

f(x)=a(x−r1)(x−r2)f(x) = a(x - r_1)(x - r_2)
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Expresses a quadratic using its roots r1r_1 and r2r_2 directly. Reveals the xx-intercepts at a glance. Available only when the quadratic has real roots.
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Asymptotes

(3 formulas)

Vertical Asymptote Rational

x=awhere Q(a)=0 and P(a)≠0x = a \quad \text{where } Q(a) = 0 \text{ and } P(a) \neq 0
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For a rational function P(x)Q(x)\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.
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Horizontal Asymptote Equal Degree

y=anbmwhen deg⁡P=deg⁡Qy = \frac{a_n}{b_m} \quad \text{when } \deg P = \deg Q
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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→±∞x \to \pm\infty. The horizontal asymptote sits at this ratio.
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Horizontal Asymptote Numerator Lower

y=0when deg⁡P<deg⁡Qy = 0 \quad \text{when } \deg P < \deg Q
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When the numerator has lower degree than the denominator, the rational function approaches zero as x→±∞x \to \pm\infty. The xx-axis itself is the horizontal asymptote.
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Rates of Change

(2 formulas)

Average Rate of Change

f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}
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Measures how much the output changes per unit of input over an interval [a,b][a, b]. Geometrically, the slope of the secant line through (a,f(a))(a, f(a)) and (b,f(b))(b, f(b)).
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Difference Quotient

f(x+h)−f(x)h\frac{f(x + h) - f(x)}{h}
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The average rate of change of ff over an interval of width hh starting at xx. Shrinking hh toward zero produces the instantaneous rate of change — the foundation of the derivative in calculus.
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Function Arithmetic
Sum of FunctionsDifference of FunctionsProduct of FunctionsQuotient of FunctionsDomain of a Combined FunctionDomain of a Quotient
Composition
Composition of FunctionsDomain of a CompositionComposition AssociativityInverse of a Composition
Inverse Functions
Inverse Function DefinitionInverse Composition PropertyInverse of an InverseDomain Range Swap of Inverse
Symmetry
Even Function TestOdd Function Test
Transformations
General TransformationVertical TranslationHorizontal TranslationVertical ReflectionHorizontal ReflectionVertical DilationHorizontal Dilation
Linear Function Forms
Slope Intercept FormPoint Slope FormStandard Form LinearSlope FormulaPerpendicular Slopes
Quadratic Function Forms
Standard Form QuadraticVertex Form QuadraticVertex from CoefficientsAxis of SymmetryFactored Form Quadratic
Asymptotes
Vertical Asymptote RationalHorizontal Asymptote Equal DegreeHorizontal Asymptote Numerator Lower
Rates of Change
Average Rate of ChangeDifference Quotient