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Functions Terms and Definitions

Core Concepts(6)
Function Families(5)
Graph Features(4)
Operations & Inverses(2)
Transformations(4)
Types & Classification(6)
AAbsolute Value FunctionFunction FamiliesAAsymptoteGraph FeaturesCComposition of FunctionsOperations & InversesCConstant FunctionFunction FamiliesDDecreasing FunctionTypes & ClassificationDDependent VariableCore ConceptsDDilationTransformationsDDomainCore ConceptsEEven FunctionTypes & ClassificationFFunctionCore ConceptsIIncreasing FunctionTypes & ClassificationIIndependent VariableCore ConceptsIInverse FunctionOperations & InversesLLinear FunctionFunction FamiliesLLocal MaximumGraph FeaturesLLocal MinimumGraph FeaturesOOdd FunctionTypes & ClassificationOOne-to-One FunctionTypes & ClassificationPParent FunctionFunction FamiliesPPiecewise FunctionTypes & ClassificationQQuadratic FunctionFunction FamiliesRRangeCore ConceptsRReflectionTransformationsRRelationCore ConceptsTTransformationTransformationsTTranslationTransformationsZZero of a FunctionGraph Features
27 of 27 terms
Core Concepts
FunctionRelationDomainRangeIndependent VariableDependent Variable
Function Families
Parent FunctionLinear FunctionQuadratic FunctionConstant FunctionAbsolute Value Function
Graph Features
Zero of a FunctionAsymptoteLocal MaximumLocal Minimum
Operations & Inverses
Composition of FunctionsInverse Function
Transformations
TransformationTranslationReflectionDilation
Types & Classification
One-to-One FunctionEven FunctionOdd FunctionIncreasing FunctionDecreasing FunctionPiecewise Function

27 terms

Core Concepts

(6 items)

Function

A rule that assigns exactly one output to each input. Formally, a function ff from set AA to set BB is a mapping f:AoBf: A o B such that every element of AA is associated with precisely one element of BB.
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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.
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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: Dom(f)={x∣f(x) is defined}\text{Dom}(f) = \{x \mid f(x) \text{ is defined}\}.
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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: Ran(f)={f(x)∣x∈extDom(f)}\text{Ran}(f) = \{f(x) \mid x \in ext{Dom}(f)\}.
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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)=x2f(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.
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intuitionnotationrelated concepts
The quantity you control. You pick its value; the function responds with an output. In y=f(x)y = f(x), the variable xx 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.
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The quantity that responds. Its value depends entirely on what was fed in. In y=f(x)y = f(x), the variable yy is dependent — it has no choice; it is whatever ff 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)impliesa=bf(a) = f(b) implies a = b.
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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)f(-x) = f(x) for all xx in its domain.
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intuitionexamplespropertiesrelated concepts
Mirror symmetry about the yy-axis. The left half of the graph is a perfect reflection of the right half. Replacing xx with −x-x changes nothing.
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Odd Function

A function satisfying f(−x)=−f(x)f(-x) = -f(x) for all xx in its domain.
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Rotational symmetry about the origin. Rotating the graph 180°180° around (0,0)(0,0) produces the same curve. Replacing xx with −x-x negates the output.
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Increasing Function

A function is increasing on an interval if a<b  ⟹  f(a)<f(b)a < b \implies f(a) < f(b) for all a,ba, b in that interval.
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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  ⟹  f(a)>f(b)a < b \implies f(a) > f(b) for all a,ba, b in that interval.
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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.
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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 ff and gg is the function (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)), where the output of gg becomes the input of ff.
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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−1f^{-1} that reverses ff: if f(a)=bf(a) = b, then f−1(b)=af^{-1}(b) = a. Equivalently, f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x.
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intuitionnotationpropertiesrelated concepts
The undo button. Whatever ff does, f−1f^{-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.
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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+bf(x) = mx + b, where mm is the slope and bb is the yy-intercept.
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A straight line. The output changes at a constant rate — every unit increase in input produces exactly mm units of change in output. No curvature, no surprises.
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Quadratic Function

A function of the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c with $a
eq 0$. The graph is a parabola.
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A U-shaped curve (or inverted U). The function changes direction at the vertex — falling then rising, or rising then falling. The leading coefficient aa controls which way the parabola opens and how wide it is.
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Constant Function

A function of the form f(x)=cf(x) = c where cc is a fixed real number. Every input produces the same output.
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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∣f(x) = |x|, returning the distance of xx from zero: ∣x∣={xif x≥0 −xif x<0|x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases}.
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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 cc where f(c)=0f(c) = 0. Also called a root of the function.
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Where the function's output hits zero — graphically, where the curve crosses or touches the xx-axis. Finding zeros means solving f(x)=0f(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).
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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)f(c) such that f(c)≥f(x)f(c) \geq f(x) for all xx in some open interval containing cc.
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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)f(c) such that f(c)leqf(x)f(c) leq f(x) for all xx in some open interval containing cc.
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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.
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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.
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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.
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A mirror image. Reflecting across the xx-axis flips the graph upside down; reflecting across the yy-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.
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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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