Functions and Polynomials - Vocabulary Flashcards (Video Notes)

A comprehensive set of vocabulary flashcards covering relations, functions, domains/ranges, composition, inverse functions, and basic polynomial concepts from the video notes.

Independent variable (X)

The input variable in a relation or function; its value determines the output.

Dependent variable (Y)

The output variable in a relation or function; its value depends on the input.

Relation

A link between elements of two sets X (inputs) and Y (outputs); a subset of ordered pairs.

Function

A special kind of relation where every element of the domain maps to exactly one element of the codomain.

Domain (Df)

Set of all inputs (X) for which a relation or function is defined.

Codomain

The set from which outputs are drawn; the target set in a function (often denoted Y).

Image

The actual output value corresponding to a specific input under a relation.

Range (Rf)

The set of all output values produced by a relation; subset of the codomain.

One-to-one function (Injective)

A function where each element of the domain maps to a unique element of the codomain (no two inputs share the same output).

Many-to-one function

A function where multiple inputs may map to the same output, but each input has exactly one output.

Not a function

A relation in which some domain element maps to more than one output; not a valid function.

Function notation (f: X → Y)

Denotes a function named f with domain X and codomain Y; f(x) is the output for input x.

Composition of functions (f ∘ g)

The function obtained by applying g first, then f: (f ∘ g)(x) = f(g(x)).

Domain of a composition (Df∘g)

The set of x in the domain of g for which g(x) lies in the domain of f.

Inverse function

If f is one-to-one (and onto between X and Y), its inverse f⁻¹ maps Y back to X such that f(f⁻¹(y)) = y and f⁻¹(f(x)) = x.

Existence of inverse

An inverse exists only if the function is one-to-one (injective).

Finding an inverse (Method 1)

Swap x and y in y = f(x), solve for x to obtain f⁻¹(y); then relabel as f⁻¹(x).

Finding an inverse (Method 2)

Let a = f⁻¹(x); express x = f(a), solve for a in terms of x, then replace a with f⁻¹(x).

Polynomial

An expression of the form an x^n + a(n−1) x^(n−1) + … + a_0 with real coefficients; degree is n.

Remainder theorem

When f(x) is divided by x − a, the remainder is f(a).

Factor theorem

x − a is a factor of f(x) if and only if f(a) = 0.

Quotient and remainder

In polynomial division, f(x) = (x − a)Q(x) + R with remainder R.

Linear function

A function of the form f(x) = ax + b with a ≠ 0; graph is a straight line.

Quadratic function

A function of the form f(x) = ax^2 + bx + c with a ≠ 0; graph is a parabola.

Cubic function

A function of the form f(x) = ax^3 + bx^2 + cx + d with a ≠ 0.

Rational function

A function of the form P(x)/Q(x) where P and Q are polynomials and Q(x) ≠ 0.

Exponential function

A function of the form a·x^k (e.g., 2^x, e^x) with exponential growth or decay.

Logarithmic function

A function of the form log_b(x) or ln(x); inverse of an exponential function.

Modulus function

Function defined by |x|, which equals x if x ≥ 0 and −x if x < 0.

Even function

A function for which f(−x) = f(x) for all x; graph is symmetric about the y-axis.

Odd function

A function for which f(−x) = −f(x) for all x; graph is symmetric about the origin.

Periodic function

A function whose values repeat at regular intervals (e.g., sine, cosine with period 2π).

Domain of f(x) = √(x−a)

x ≥ a; the expression under the square root must be nonnegative.

Domain of f(x) = 1/(x−a)

x ≠ a; division by zero is not allowed.

Domain of f(x) = √(5 − 2x)

x ≤ 2.5; requires 5 − 2x ≥ 0 for the square root to be real.

Composite inverse relationships

If two functions are inverses, applying them in either order yields the identity function on the respective domain.

Inverse notation

f⁻¹(x) is the inverse function of f; it reverses the mapping of f.

Domain and range examples (contextual terms)

Examples include: Domain of f(x) = √(x−2) is x ≥ 2; Range is y ≥ 0; Domain of f(x) = x^2 is all real x; Range is y ≥ 0.

Function notation for domain/range

Df is the set of inputs (domain); Rf is the set of outputs (range) for a function f.

Notation for composition domains

Df∘g = { x | x ∈ Dg and g(x) ∈ Df }; Dg∘f = { x | x ∈ Df and f(x) ∈ Dg }.

Polynomial factorization

Expressing a polynomial as a product of factors, e.g., f(x) = (x − a)·Q(x) when a is a root.