additive transformation

A transformation of a function involving addition or subtraction, resulting in vertical and horizontal translations.

analytic representation

An algebraic formula or equation that explicitly defines a function.

commutative

A property where the order of operations does not affect the result; composition of functions is not commutative, meaning f ∘ g typically produces a different result than g ∘ f.

composite function

A function formed by combining two or more functions, where the output of one function becomes the input of another function, denoted as f ∘ g or f(g(x)).

composition of functions

A function operation where one function is applied to the output of another function, written as (f ∘ g)(x) = f(g(x)).

decomposed

The process of breaking down a complex function into simpler component functions that can be composed together.

domain

The set of all possible input values for which a function is defined.

function composition

The process of combining two or more functions where the output of one function becomes the input of another function.

graphical representation

A visual representation of a function displayed on a coordinate plane.

horizontal dilation

A transformation that stretches or compresses the graph of a function horizontally by multiplying the input by a constant factor b, written as g(x) = f(bx).

horizontal translation

A transformation that shifts the graph of a function left or right by adding a constant to the input, written as g(x) = f(x + h).

identity function

The function h(x) = x, which serves as the line of reflection between inverse exponential and logarithmic functions.

input value

The x-values or independent variable values used as inputs to a function.

multiplicative transformation

A transformation of a function involving multiplication, resulting in vertical and horizontal dilations.

numerical representation

A representation of a function using tables of values or ordered pairs (x, y).

output value

The y-values or results produced by a function for given input values.

vertical dilation

A transformation that stretches or compresses the graph of a function vertically by multiplying the function by a constant factor a, written as g(x) = af(x).

vertical translation

A transformation that shifts the graph of a function up or down by adding a constant k to the function, written as g(x) = f(x) + k.