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A cubic function is a type of polynomial that features a variable raised to the third power as its highest degree. In mathematical terms, a general cubic function can be written as
\( f(x) = ax^{3} + bx^{2} + cx + d \)
where \( a, b, c \), and \( d \) are constants and the highest exponent of variable \( x \) is 3, thus making it a 'cubic' term. The graph of a cubic function is known as a 'cubic curve', and it can display a wide variety of shapes. However, all cubic functions possess certain characteristics, such as having at least one point of inflection and possibly multiple real roots.
In the given exercise, the cubic function presented is a special case where the only term is \( y^{3}=x^{3} \). This particular cubic function represents a situation where each variable influences the outcome equally, simplifying to the linear relationship \( y = x \) after performing the appropriate operations, as shown in the 'Steps' provided.

Cube root operations are essential when dealing with cubic functions. The cube root of a number \( a \) is a value \( b \) that, when multiplied by itself three times, equals \( a \), expressed as \( b^{3} = a \) or \( b = \sqrt[3]{a} \). This is the inverse operation of cubing a number.
In the context of isolating \( y \) in a cubic function, applying the cube root to both sides of an equation like \( y^{3} = x^{3} \) allows us to find the original value of \( y \) before it was cubed. As shown in the exercise, performing the cube root operation thus simplifies \( \sqrt[3]{y^{3}} \) to \( y \) and \( \sqrt[3]{x^{3}} \) to \( x \). Understanding this operation is vital to work with cubic equations and to isolate variables effectively.

Simplifying equations is a crucial skill in algebra. It involves reducing equations to a more basic or more easily solvable form, often by performing operations that cancel out or combine terms. The aim is to isolate the variable of interest, in this case, \( y \), as a function of another variable, \( x \).
The exercise we're looking at serves as an example of simplifying an equation through the process of solving a cubic equation. To simplify \( y^{3} = x^{3} \), we apply the cube root, because it is the inverse operation to cubing. This process is essentially 'undoing' the operation that has been applied to \( y \), revealing its relationship to \( x \). By carrying out these steps, we move from a more complex cubic equation to a simple linear one, exemplifying the goal of simplification in algebra.
Understanding these principles enables students to tackle more complicated algebraic problems by breaking them down into simpler steps to isolate and solve for variables.