{"id":7620,"date":"2022-04-15T17:47:46","date_gmt":"2022-04-15T12:17:46","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=7620"},"modified":"2025-06-02T20:50:24","modified_gmt":"2025-06-02T15:20:24","slug":"what-is-a-function","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-a-function\/","title":{"rendered":"What is a function?"},"content":{"rendered":"\n

Definition: Let X and Y are two sets. A relation f from X to Y is a function<\/a> if for every x \u2208 X there is a unique element y \u2208 Y, such that (x, y) \u2208 X<\/p>\n\n\n\n

Here we denote it as f: X \u2192 Y, which means f is a function from X to Y.<\/p>\n\n\n\n

Hence, We represent it as-<\/p>\n\n\n\n

\"This<\/figure>\n\n\n\n

Example: Let \"This And \"This And \"This<\/p>\n\n\n\n

Hence we represent the function as below<\/p>\n\n\n\n

\"This<\/figure>\n\n\n\n

Here A is the domain of f and B is the co-domain of f<\/p>\n\n\n\n

Surjective function (Onto-Mapping)<\/strong><\/p>\n\n\n\n

A mapping f: X \u2192 Y said to be onto-mapping if the range set .<\/p>\n\n\n\n

If  f: X \u2192 Y is onto then each element of B if f-image of at least one element of X. <\/p>\n\n\n\n

That means,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

A mapping is a into if it is not onto.<\/p>\n\n\n\n

Injective <\/strong>fn (One-To-One Mapping)-<\/strong><\/h2>\n\n\n\n

A mapping f: X \u2192 Y is said to be one-to-one mapping if distinct elements of X are mapped into distinct elements of Y.<\/p>\n\n\n\n

That means if f is a one-to-one mapping, then-<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Or in other words,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Bijective functions (One-To-One, Onto)-<\/strong><\/p>\n\n\n\n

A mapping f: X \u2192 Y is the bijective mapping if it is both one-to-one and onto. <\/p>\n\n\n\n

Identity function (Identity Mapping)-<\/strong><\/p>\n\n\n\n

If f: X \u2192 Y  is a function such that every element of X is mapped onto itself then f is said to be an identity mapping.<\/p>\n\n\n\n

We denote identity mapping by <\/p>\n\n\n\n

If is an identity mapping.<\/p>\n\n\n\n

Partial functions-<\/strong><\/p>\n\n\n\n

Let X and Y be sets and A be a subset of X. A function f from A to Y is called a partial func. from X to Y. The set A is a called the domain of f.<\/p>\n\n\n\n

Constant function-<\/strong><\/p>\n\n\n\n

Let f: X \u2192 Y  then f is a constant function if every element of X is map onto the same element of Y.<\/p>\n\n\n\n

Inverse functions-<\/strong><\/p>\n\n\n\n

Let f: X \u2192 Y is a one-one and onto mapping, then is called the inverse mapping of f.<\/p>\n\n\n\n

We define the Inverse mapping as<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Composition of Functions<\/strong><\/p>\n\n\n\n

Let f: X \u2192 Y and g: Y \u2192 Z are two mappings, then we denote the composition of these two mappings f and g by gof is the mapping from X into Z.<\/p>\n\n\n\n

Hence, we define it as<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

That means  is a mapping defined as below-<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Invertible function-<\/strong><\/p>\n\n\n\n

A function f from a set X to a set Y is an invertible func if there exists a function g from Y to X, such that-<\/p>\n\n\n\n

f(g(y)) = y And g(f(x)) = x For every x \u2208 X and y\u2208 Y<\/p>\n\n\n\n

Recursive Functions- <\/strong>A function f is a Primitive recursive if we obtain it from the initial functions by a finite number of operations of composition and recursion.<\/p>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n