Determine whether the given functions are inverses. ƒ= {(2,5), (3,5), (4,5)}; g = {(5,2)}

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Recall that two functions \( f \) and \( g \) are inverses if and only if \( f(g(x)) = x \) for every \( x \) in the domain of \( g \), and \( g(f(x)) = x \) for every \( x \) in the domain of \( f \).

Identify the domain and range of each function from the given sets: \( f = \{(2,5), (3,5), (4,5)\} \) and \( g = \{(5,2)\} \).

Check \( f(g(x)) \): For each \( x \) in the domain of \( g \), find \( g(x) \) and then apply \( f \) to that result. Here, \( g(5) = 2 \), so calculate \( f(2) \).

Check \( g(f(x)) \): For each \( x \) in the domain of \( f \), find \( f(x) \) and then apply \( g \) to that result. For example, \( f(2) = 5 \), so calculate \( g(5) \). Repeat for \( x = 3 \) and \( x = 4 \).

Compare the results from steps 3 and 4 to the original inputs. If \( f(g(x)) = x \) and \( g(f(x)) = x \) hold for all relevant \( x \), then \( f \) and \( g \) are inverses; otherwise, they are not.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definition of Inverse Functions

Two functions f and g are inverses if applying one after the other returns the original input, meaning f(g(x)) = x and g(f(x)) = x for all x in their domains. This implies that the output of f becomes the input of g and vice versa, effectively reversing each other's operations.

Function and Relation Notation

Functions can be represented as sets of ordered pairs, where each input (first element) maps to exactly one output (second element). Understanding this notation helps analyze whether the pairs in f and g correspond correctly to satisfy the inverse relationship.

One-to-One Functions and Invertibility

A function must be one-to-one (injective) to have an inverse function, meaning no two different inputs share the same output. If a function is not one-to-one, its inverse is not well-defined as a function, which is crucial when checking if f and g are inverses.