Function

In mathematics, a function is a rule or relation between two non-empty sets that assigns each element of the first set to exactly one element of the second set. It is commonly denoted as f: A → B. In this notation, set A is the domain of the function and set B is the codomain. The set of all images of elements from the domain under the function is called the range, denoted by f(A). The range is a subset of the codomain (f(A) ⊆ B). Functions are classified into various types based on their outputs, the relationship between their domain and range, and their algebraic properties. This classification plays a fundamental role in mathematical analysis and problem solving.

One-to-One Function (Injective Function)

A function is called one-to-one if each distinct element in the domain maps to a distinct image in the codomain. Mathematically, for a function f: A → B, if for all x₁, x₂ ∈ A, the condition x₁ ≠ x₂ implies f(x₁) ≠ f(x₂), or equivalently, if f(x₁) = f(x₂) implies x₁ = x₂, then f is one-to-one. This means that each element in the codomain is associated with at most one element from the domain.

To determine whether a function is one-to-one from its graph, the horizontal line test is applied. If every horizontal line drawn parallel to the x-axis intersects the graph at most once, the function is one-to-one. If any horizontal line intersects the graph at more than one point, the function is not one-to-one.

Example: The function f: ℝ → ℝ, f(x) = 2x + 3 is one-to-one because for any two different input values (e.g., x₁ = 1, x₂ = 2), the outputs are always different (f(1) = 5, f(2) = 7).

Onto Function (Surjective Function)

A function is called onto if every element in the codomain is the image of at least one element in the domain; that is, no element in the codomain is left unmapped. In other words, if the range of the function equals the codomain (f(A) = B), then the function is onto. This means that every element in the codomain corresponds to the image of at least one element in the domain.

The horizontal line test is not used to determine surjectivity; surjectivity is determined by whether the range equals the codomain. If every horizontal line drawn parallel to the x-axis within the codomain’s range intersects the graph at least once, the function is onto. If there exists any horizontal line that does not intersect the graph, the function is not onto.

Example: The function f: ℝ → ℝ, f(x) = x³ is onto because for every real number y in the codomain (ℝ), there exists a real number x (x = ³√y) in the domain (ℝ) such that y = x³.

Into Function

A function that is not onto is called an into function. For a function f: A → B, if at least one element in the codomain has no preimage in the domain, then the function is into. In this case, the range is a proper subset of the codomain (f(A) ≠ B). A function is either onto or into.

Example: The function f: ℝ → ℝ, f(x) = x² is an into function because although the codomain is all real numbers (ℝ), the range consists only of non-negative real numbers ([0, ∞)). Negative numbers in the codomain (e.g., -1) have no preimage.

One-to-One and Onto Function (Bijective Function)

A function that is both one-to-one and onto is called a bijective function. In such functions, each element of the domain corresponds to exactly one distinct element in the codomain, and no element in the codomain is left unmapped. Bijective functions establish a perfect one-to-one correspondence between the domain and codomain. These functions are invertible and are also known as permutation functions.

Identity Function

A function that maps every element of its domain to itself is called the identity function and is usually denoted by I. If A is a non-empty set, then I: A → A is defined by I(x) = x. In the identity function, the image of any element is always the element itself. The graph of the function f(x) = x is the line y = x, known as the first angle bisector in the coordinate plane.

Example: If f(x) is the identity function, then f(5) = 5 and f(3x - 2) = 3x - 2.

Constant Function

A function that maps all elements of its domain to a single element in the codomain is called a constant function. For a function f: A → B, if there exists a constant c ∈ B such that f(x) = c for all x ∈ A, then f is a constant function. The rule of this function is independent of the variable x. Its graph is a horizontal line parallel to the x-axis.

Example: The function f: ℝ → ℝ, f(x) = 7 is a constant function because for all values of x in the domain, the output is always 7. f(1) = 7, f(-100) = 7.

Zero Function

The zero function is a special case of a constant function in which every element of the domain is mapped to the element 0 in the codomain. That is, f(x) = 0.

Linear Function

A function f: ℝ → ℝ defined by f(x) = ax + b, where a and b are real numbers and a ≠ 0, is called a linear function. The graph of such functions is a straight line in the Cartesian plane. The coefficient a represents the slope of the line, and b represents the y-intercept.

Example: f(x) = 3x - 5 is a linear function. Its graph is a straight line with slope 3 and y-intercept -5.

Piecewise Function

A function defined by different rules on disjoint subintervals of its domain is called a piecewise function. The points where the rule changes are called critical points. To evaluate a piecewise function at a given point, determine which subinterval the point belongs to and apply the corresponding rule.

Example: The function f(x) = { x+2, x < 0; x², 0 ≤ x < 5; 4x, x ≥ 5 } is a piecewise function. To find f(-3), use the rule for x < 0: f(-3) = -3 + 2 = -1. To find f(4), use the rule for 0 ≤ x < 5: f(4) = 4² = 16.

Odd and Even Functions

Functions can be classified as odd or even based on their symmetry properties. This classification typically applies to functions whose domain is symmetric about the origin (i.e., if x ∈ A, then -x ∈ A).

Even Function

A function for which f(-x) = f(x) holds for all x ∈ A is called an even function. The graphs of even functions are symmetric with respect to the y-axis.

Example: f(x) = x² + 1 is an even function because f(-x) = (-x)² + 1 = x² + 1 = f(x).

Odd Function

A function for which f(-x) = -f(x) holds for all x ∈ A is called an odd function. The graphs of odd functions are symmetric with respect to the origin.

Example: f(x) = x³ is an odd function because f(-x) = (-x)³ = -x³ = -f(x).

Equal Functions

Two functions are called equal if they have the same domain and codomain and produce the same output for every element in the domain. For functions f: A → B and g: A → B, if f(x) = g(x) for all x ∈ A, then f and g are equal (f = g).

Example: Let A = {-1, 1}. The functions f: A → ℝ, f(x) = x³ and g: A → ℝ, g(x) = x are equal because f(-1) = -1, g(-1) = -1 and f(1) = 1, g(1) = 1.

Composite Function

Given non-empty sets A, B, C and functions f: A → B and g: B → C, the composite function of f and g, denoted by (g o f), is a new function from A to C defined by (g o f)(x) = g(f(x)). In this process, f is applied first, and its output is used as the input to g.

Example: If f(x) = x + 2 and g(x) = 3x, then (g o f)(x) = g(f(x)) = g(x + 2) = 3(x + 2) = 3x + 6.

Inverse Function

The inverse function of a bijective function f: A → B is denoted by f⁻¹: B → A and maps each element of the codomain back to its original preimage in the domain. If y = f(x), then x = f⁻¹(y). For the inverse of a function to also be a function, the original function must be bijective. To find the inverse, solve the equation y = f(x) for x, then replace x with f⁻¹(x) and y with x.

Periodic Function

A function whose graph and values repeat at regular intervals along the x-axis is called a periodic function. If T is a non-zero constant such that f(x + T) = f(x) for all x in the domain, the smallest positive such T is called the period of the function. Trigonometric functions (sine, cosine, etc.) are the most well-known examples of periodic functions.