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Inverse functions are functions that reverse the effect of the original functions. If you have a function that takes an input and produces an output, the inverse function takes the output back to the original input.
An important aspect of inverse functions is that not all functions have inverses. For a function to have an inverse, it must be one-to-one. This means each output is produced by exactly one input.
To find an inverse, follow these steps:

• Replace the function notation (usually f(x)) with y.
• Swap the roles of x and y.
• Solve for y again.

The new y will be your inverse function. An example of the process is shown in the provided solution.

Solving for y is the process of isolating the variable y on one side of an equation. This typically involves using algebraic operations such as addition, subtraction, multiplication, division, and sometimes more complex functions like roots and exponents.
In the provided example, we start with the equation:

$$ x^3 y = -5 $$

To solve for y in this context, we divide both sides by x^3 to isolate y:

$$ y = \frac{-5}{x^3} $$

After finding y, we can proceed to determine the inverse by swapping x and y and then isolating the new y again. These steps ensure that the relationship described by the original function can be reversed.

The cube root of a number is the value that, when multiplied by itself three times, gives the original number. Mathematically, the cube root of a number a is written as:

\(\frac{-5}{x} \)

and is typically denoted as \( \text{root[3]{a}} \)
Cube roots are the inverse operation to cubing a number, just as square roots are the inverse operation to squaring a number. In the context of the given problem, after isolating \( \frac{-5}{x} \), we take the cube root of both sides to solve for the new y:

\( y = \text{root[3]{-5/x}} \)

Understanding cube roots is essential when dealing with polynomial equations, especially those involving cubes. Mastering this concept is a key step in solving equations that require you to undo exponents and find inverse relationships.