Question

Find the inverse of the functions. $f(x)=x^{3}+5$

   Find the inverse of the functions.
$f(x)=x^{3}+5$

Precalculus

Jay Abramson 1st Edition

Chapter 3, Problem 555

Instant Answer

Solved by Expert Amrita Bhasin

02/25/2020

Step 1

Step 1: First, we start with the given function $f(x)=x^{3}+5$. Show more…

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Find the inverse of the functions. $f(x)=x^{3}+5$

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Key Concepts

Domain and Range Swap

When determining the inverse function, the domain and range of the original function are interchanged. This concept is crucial because the set of possible inputs for the inverse is the set of outputs of the original function, and vice versa, which helps ensure that the inverse function is properly defined.

Algebraic Manipulation

Finding an inverse typically involves algebraically solving the equation y = f(x) for x in terms of y. This process includes operations such as isolating terms, performing inverse operations like taking roots or logarithms, and then interchanging the variables, which is essential for deriving the inverse function.

Inverse Function

An inverse function essentially 'undoes' what the original function does. If you apply the function and then its inverse, you return to the original input. This concept is fundamental because it allows solving for input values in terms of output values and understanding the reversibility of operations.

One-to-One Function

A function must be one-to-one to have an inverse. This means that every output is produced by exactly one input. The one-to-one property ensures that the inverse is also a well-defined function because it guarantees that each element in the range of the original function corresponds to a unique element in its domain.

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