Use Theorem 5 to determine if g is the inverse of f. f(x)=(2 x-3)/(x+4) ; g(x)=(3+4 x)/(2-x)

Name: Problem 49, Chapter 1: Precalculus: Graphs and Models
Uploaded: 2020-07-03T18:27:35-08:00
Duration: 3 min 6 s
Channel: Charles Carter
Description: Use Theorem 5 to determine if g is the inverse of f. f(x)=(2 x-3)/(x+4) ; g(x)=(3+4 x)/(2-x)

step 1 Okay, so this problem, the first step to using the theorem 5 is to show is to find the compositi

Use Theorem 5 to determine if g is the inverse of f. f(x)=(2...

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Key Concepts

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f has an inverse g, then 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. This mutual cancellation property is the defining characteristic of inverse functions.

Function Composition

Function composition involves applying one function to the result of another. In the context of inverse functions, composing a function with its purported inverse should yield the identity function. That is, if g is the inverse of f, then the compositions f(g(x)) and g(f(x)) must both return the original input x.

Verifying Inverse Relationships Using Theorems

Theorem 5 (or similar theorems) typically provides sufficient conditions for determining if one function is the inverse of another by checking if the compositions of the functions yield the identity function. This approach streamlines the verification of inverse relationships.

Domain Considerations

When working with inverses, it is crucial to be mindful of the domains of the functions involved. The inverse function must have a domain and range that correspond appropriately to those of the original function. Any restrictions on x values for which the functions are defined must be taken into account when verifying the inverse relationship.

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