For each pair of functions, find (f^-1 ∘f)(x) f(x)=(x+1)/(x-2) and f^-1(x)=(2 x+1)/(x-1)

Name: Problem 83, Chapter 9: Algebra for College Students
Uploaded: 2020-07-06T03:44:33-08:00
Duration: 1 min 4 s
Channel: Ankit Gupta
Description: For each pair of functions, find (f^-1 ∘f)(x) f(x)=(x+1)/(x-2) and f^-1(x)=(2 x+1)/(x-1)

step 1 F of X is equal to Y is equal to 2x raised to the power 3. Now switching the values of X and Y,

For each pair of functions, find (f^-1 ∘f)(x)...

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Key Concepts

Function Composition

Function composition is the process of applying one function to the results of another function. In the context of (f?¹ ? f)(x), this means that you first apply the function f to x, and then you apply the inverse function f?¹ to the result. This composition is designed so that the output will recover the original input x, provided that x is within the appropriate domain.

Inverse Functions

An inverse function essentially reverses the effect of the original function. For a function f, the inverse function f?¹ satisfies the equation f?¹(f(x)) = x for every x in the domain of f. This means that applying f and then its inverse f?¹ brings you back to the initial value, demonstrating an important property of invertible functions.

Domain Restrictions

Domain restrictions refer to the values for which the functions f and f?¹ are defined, typically determined by avoiding division by zero or other undefined operations. When working with rational functions like these, it is crucial to consider the values of x that would cause the denominator to be zero, as these values are excluded from the domain of the function. This also affects the validity of the composition (f?¹ ? f)(x).

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