Question

Find the inverse of the function. $$f(x)=3 x^{3}-4$$

   Find the inverse of the function.
$$f(x)=3 x^{3}-4$$

Precalculus

Jay Abramson 1st Edition

Chapter 3, Problem 749

Instant Answer

Solved by Expert Albert T.

06/14/2020

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So, we have: $$y=3x^{3}-4$$ Show more…

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Find the inverse of the function. $$f(x)=3 x^{3}-4$$

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Key Concepts

One-to-One Function

A one-to-one function is a function in which each element of the domain maps to a unique element of the range, ensuring that no two distinct inputs produce the same output. This property is essential for the existence of an inverse function, as it guarantees that each output of the function corresponds to exactly one input.

Algebraic Manipulation

Algebraic manipulation involves performing operations such as addition, subtraction, multiplication, division, and applying inverse operations (like taking cube roots) to both sides of an equation in order to isolate a variable. This step-by-step process is fundamental in finding the inverse of a function by rewriting the equation to express x in terms of y.

Inverse Function

An inverse function essentially reverses the operation of the original function. For a function f, its inverse f?¹ satisfies the property that f?¹(f(x)) = x for all x in the domain, meaning that applying the function and then its inverse returns the original value. This concept is crucial for solving equations where you need to ‘undo’ a transformation.

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