Find a formula for f^-1(x) . Identify the domain and range of f^-1. Verify that f and f^-1 are i

step 1 We have the function, f of x, equals 2 over x minus 1. We want to find the inverse function. Let

Find a formula for f^-1(x) . Identify the domain and range...

Key Concepts

Function Composition

Function composition is the application of one function to the results of another. In the context of inverses, composing a function with its inverse tests whether the combination returns the initial input. This operation underpins the verification step for inverse functions by demonstrating that f(f?¹(x)) = x and f?¹(f(x)) = x.

Inverse Functions

An inverse function essentially reverses the effect of the original function. For a function f, its inverse f?¹ satisfies the condition that applying f to f?¹ (or vice versa) returns the original input. To find an inverse algebraically, one typically replaces f(x) with y, interchanges x and y, and then solves for y, which becomes f?¹(x).

Domain and Range

The domain of a function consists of all the input values for which the function is defined, while the range consists of all possible outputs. When determining the inverse of a function, the domain and range effectively switch roles. This transformation is crucial in verifying that the inverse function has been correctly determined, ensuring that outputs of the original function become valid inputs in the inverse.

Verification of Inverse Functions

Verifying that two functions are inverses involves composing them in both orders (f(f?¹(x)) and f?¹(f(x))) and confirming that the result is the identity function; that is, it returns the input value x for all x in the appropriate domains. This process establishes that each function precisely 'undoes' the effect of the other.

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