Functions and Relations

The domain of a function is the set of all possible values of the independent variable for which the function is defined. Since the square root function is defined only for nonnegative real numbers, the domain of (f(x) = \sqrt{x}) is all nonnegative real numbers.

The range of a function is the set of all possible values of the dependent variable. Since the square function always produces a nonnegative real number, the range of (f(x) = x^2) is all nonnegative real numbers.

A function is injective if it preserves distinct elements, meaning that if (x_1 \neq x_2), then (f(x_1) \neq f(x_2)). The function (f(x) = x^3) is injective because (x_1^3 \neq x_2^3) whenever (x_1 \neq x_2).

A function is surjective if it maps every element of the codomain to at least one element of the domain. The function (f(x) = \sin(x)) is surjective because for every real number (y) in the codomain, there exists a real number (x) in the domain such that (f(x) = y).

A function is bijective if it is both injective and surjective. The function (f(x) = \tan(x)) is bijective because it is both injective ((x_1 \neq x_2) implies (\tan(x_1) \neq \tan(x_2))) and surjective (for every real number (y) in the codomain, there exists a real number (x) in the domain such that (\tan(x) = y)).

The inverse of a function (f(x)) is the function (f^{-1}(x)) such that (f(f^{-1}(x)) = x) and (f^{-1}(f(x)) = x). To find the inverse of (f(x) = 2x + 1), we can solve for (x) in terms of (f(x)): (f(x) = 2x + 1) (\Rightarrow 2x = f(x) - 1) (\Rightarrow x = \frac{f(x) - 1}{2}) (\therefore f^{-1}(x) = \frac{x - 1}{2}).

The composition of two functions (f(x)) and (g(x)) is the function ((f \circ g)(x) = f(g(x))). In this case, ((f \circ g)(x) = f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1).

A relation is a function if each element of the domain is paired with exactly one element of the codomain. In other words, a relation is a function if there are no two different elements of the codomain that are paired with the same element of the domain. The relation {(1, 2), (2, 3), (3, 4), (4, 5)} is a function because each element of the domain (1, 2, 3, and 4) is paired with exactly one element of the codomain (2, 3, 4, and 5, respectively).

A relation is reflexive if every element of the domain is related to itself. In other words, a relation is reflexive if ((a, a)) is in the relation for every (a) in the domain. The relation {(1, 1), (2, 2), (3, 3), (4, 4)} is reflexive because ((1, 1), (2, 2), (3, 3),) and ((4, 4)) are all in the relation.

A relation is symmetric if whenever ((a, b)) is in the relation, then ((b, a)) is also in the relation. In other words, a relation is symmetric if it is unchanged when the order of the elements in each pair is reversed. The relation {(1, 2), (2, 1), (3, 4), (4, 3)} is symmetric because ((1, 2)) and ((2, 1)), ((3, 4)) and ((4, 3)) are all in the relation.

A relation is transitive if whenever ((a, b)) and ((b, c)) are in the relation, then ((a, c)) is also in the relation. In other words, a relation is transitive if it is possible to go from any element of the domain to any other element of the domain by following the relation. The relation {(1, 2), (2, 3), (3, 4), (4, 5)} is transitive because ((1, 2), (2, 3),) and ((3, 4)) are all in the relation, so ((1, 4)) is also in the relation.

An equivalence relation is a relation that is reflexive, symmetric, and transitive. The relation {(1, 1), (2, 2), (3, 3), (4, 4)} is an equivalence relation because it is reflexive (((a, a)) is in the relation for every (a) in the domain), symmetric (whenever ((a, b)) is in the relation, then ((b, a)) is also in the relation), and transitive (whenever ((a, b)) and ((b, c)) are in the relation, then ((a, c)) is also in the relation).

The domain of a relation is the set of all possible values of the independent variable. In this case, the independent variable is (x), so the domain of the relation (R = {(x, y) | x + y = 5) is the set of all real numbers, (\mathbb{R}).

The range of a relation is the set of all possible values of the dependent variable. In this case, the dependent variable is (y), so the range of the relation (R = {(x, y) | x + y = 5) is the set of all real numbers, (\mathbb{R}).

A relation is a function if each element of the domain is paired with exactly one element of the codomain. In this case, each value of (x) in the domain is paired with exactly one value of (y) in the codomain, so the relation (R = {(x, y) | x + y = 5) is a function.