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In mathematics, a function is a relation between sets that associates to every element of a first set exactly one element of the second set. Think of it as a machine where you input a number, and the function provides exactly one output. For a relation to be a function, every value of the independent variable, often represented by x, must be associated with one and only one value of the dependent variable, typically y. When we look at the equation from our exercise, \( x + y^3 = 27 \), the goal is to determine if y can be uniquely determined for each x.

Following our solution steps, we first isolate y to see the potential function more clearly. Any predictable process that helps us to find y based on a specific x, such as the manipulation done in Step 1, is part of defining y as a function of x. It's crucial because it shows whether each input x has a single output y.

The cube root extraction is essential in solving equations where the variable is raised to the third power, as seen in our exercise. When y is cubed (\(y^3\)), we reverse this by taking the cube root of both sides to solve for y. The cube root of a number is a value that, when cubed, gives the original number. For example, the cube root of 27 is 3 because \(3^3 = 27\).

In Step 2 of our solution, \(cube root of (27 - x)\) was taken to isolate y. This process illustrates a functional relationship because, for any real number x, \(cube root of (27 - x)\) will yield a real number y. This operation is crucial as it demonstrates that our equation can produce a single y for each x, aligning with the definition of a function.

The vertical line test is a visual way to determine if a curve in the coordinate plane is the graph of a function. This test states that if a vertical line intersects a curve at more than one point, then the curve does not represent a function because that would mean a single input (x-value) is mapping to multiple outputs (y-values), which contradicts the function definition. However, if each vertical line crosses the graph at most once, every x corresponds to one y, and we indeed have a function.

In Step 3, although we don't have a literal graph to test, the equation after cube root extraction guarantees that each x will produce one unique y, passing the vertical line test conceptually. Whenever you're given an equation and asked if it defines y as a function of x, thinking about the vertical line test can help you discern if there's a one-to-one relationship between x and y.