Use the definition of inverses to determine whether f and g are inverses. f(x)=(2)/(x+6), g(x)

step 1 Here we have two functions f of f and g and we need to determine that the f and g are inverse to

Use the definition of inverses to determine whether f and g...

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Key Concepts

Identity Function

The identity function is a function that returns its input unchanged. In verifying inverse functions, showing that the composition of the function with its potential inverse yields the identity function is key. That is, if f(g(x)) = x and g(f(x)) = x, then the functions are inverses.

Domain Considerations

When determining if two functions are inverses, it is important to consider their domains and ranges. The domains of the function and its inverse must be such that each function is defined for the necessary input values, ensuring that the compositions are valid and yield the identity function.

Function Composition

Function composition involves applying one function to the result of another. In the context of inverses, composing a function with its inverse should result in the identity function, which confirms that the two functions cancel each other’s effects.

Inverse Functions

An inverse function essentially reverses the effect of the original function. For a function f and its inverse f?¹, applying f then f?¹ (or vice versa) returns the original input. This property is expressed as f(f?¹(x)) = x and f?¹(f(x)) = x for all x in the domains of f and f?¹, respectively.

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