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Understanding function definitions is crucial. A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range), where each input is related to exactly one output.
In mathematical terms, if we say there is a function **f** that maps each element in set **X** (the domain) to an element in set **Y** (the range), we can express this as:
\[ f: X \rightarrow Y \] The function f contains ordered pairs, where each pair is written as \( (a, b) \), meaning the function maps input **a** to output **b**.
For example, in the given functions, **f** and **g**:
\[ f = \{(1,1),(3,3),(5,5)\} \] \[ g = \{(1,1),(3,3),(5,5)\} \] The function **f** maps 1 to 1, 3 to 3, and 5 to 5. The same applies to **g**.

Inverse functions reverse the roles of inputs and outputs. If you have a function **f** that maps **x** to **y**, then the inverse function, denoted as **f\textsuperscript{-1}**, maps **y** back to **x**.
In mathematical notation, if \[ f(x) = y \] then \[ f^{-1}(y) = x \] For two functions **f** and **g** to be inverses of each other, they must satisfy the conditions:

• **f**(*x*) = *y*
• **g**(*y*) = *x*

In the given functions:
\[ f = \{(1,1),(3,3),(5,5)\} \] \[ g = \{(1,1),(3,3),(5,5)\} \] each element maps to itself; hence, **f** is its own inverse. This means **f** and **g** are inverses of each other, as they map their domains and ranges in reverse.

Ordered pairs are a fundamental concept when working with functions and their inverses. An ordered pair is written as (a, b), where:

• The first element (a) is from the domain.
• The second element (b) is from the range.

In the context of inverse functions, the ordered pairs (a, b) of function **f** and (b, a) of function **g** should match.
Let's explore the ordered pairs in the given functions **f** and **g**:
Function **f** has ordered pairs:

• (1, 1)
• (3, 3)
• (5, 5)

Function **g** has the same ordered pairs:

• (1, 1)
• (3, 3)
• (5, 5)

Since each pair (a, b) in **f** corresponds to (b, a) in **g**, the functions are inverses. Thus, the inversion criterion is satisfied, confirming their inverse relationship.