Functions The Algebra of Functions (MA137) – Vocabulary Flashcards

Vocabulary flashcards covering key concepts from the lecture notes on domains, graphs, tests, inverses, and composition of functions.

Domain of a function

The set of inputs x for which f(x) is defined (the input values the function can accept).

Codomain

The set B in f: A → B; the target set that contains all possible outputs of the function.

Range

The set {f(x) | x ∈ A}, i.e., all actual outputs the function produces from its domain.

Graph of a function

The set of points (x, f(x)) in the xy-plane; the graph of y = f(x).

Vertical Line Test

A curve is the graph of a function if and only if no vertical line intersects the curve more than once.

Horizontal Line Test

A function is one-to-one (injective) if and only if no horizontal line intersects its graph more than once.

Notation f: A → B, x ↦ f(x)

A concise way to express a function as a mapping from domain A to codomain B, sending x to f(x).

One-to-one function (injective)

A function where distinct inputs have distinct outputs; f(x1) = f(x2) implies x1 = x2.

Inverse function

For a one-to-one function f: A → B, the inverse f^{-1}: B → A satisfies f^{-1}(y) = x iff f(x) = y.

f^{-1} vs 1/f

f^{-1} denotes the inverse function, not the reciprocal 1/f; they are generally different concepts.

Properties of inverse functions

f^{-1}(f(x)) = x for all x in A and f(f^{-1}(y)) = y for all y in B.

Domain and range of inverse

The inverse f^{-1} has domain B and range A.

How to find the inverse (procedure)

1) write y = f(x); 2) solve for x in terms of y; 3) interchange x and y to get the inverse.

Graph of the inverse function

The graph of f^{-1} is the reflection of the graph of f across the line y = x.

Composition of functions

(f ∘ g)(x) = f(g(x)); the composition is defined where g(x) lies in the domain of f.

Non-commutativity of composition

In general, f ∘ g ≠ g ∘ f.

Domain considerations for combined functions

For (f+g), (f−g), (fg), (f/g), the domain is A ∩ B, and for f/g also require g(x) ≠ 0.

Equality of functions

Two functions f and g are equal if they have the same domain and f(x) = g(x) for all x in that domain.

Graph reading convention

The value f(x) is read as the height of the graph at the input x.

Equation form and graph

The graph corresponds to the equation y = f(x); its points are all pairs (x, y) with y = f(x).