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Cubic equations are mathematical expressions where the highest power of the variable is three. This makes them a type of polynomial equation classified as degree three. An example of a simple cubic equation is \(y^3 = 27 - x\). In such equations, you often see terms such as \(x^3\), \(ax^2\), and \(bx\), where the coefficients \(a\) and \(b\) can be any real numbers.

Cubic equations can have one, two, or three real roots. The nature of these roots can be complex or real. Unlike quadratic equations that might sometimes not have real roots, cubic equations always guarantee at least one real root. Solving a cubic equation generally involves isolating the cubic term and finding its roots, usually by factoring or using a radical expression like the cubic root.

Solving equations is all about finding the value of the variable that makes the equation true. In the exercise given, we need to isolate \(y\) to express it as a function of \(x\). This is done by rearranging the equation to \(y^3 = 27 - x\). To find \(y\), you solve it by taking the cubic root of both sides, yielding \(y = \sqrt[3]{27 - x}\).

The process of solving equations typically involves:

• Rearranging the terms — to get the variable of interest by itself.
• Applying mathematical operations — such as addition, subtraction, multiplication, division, or extracting roots.
• Checking your solutions — ensure the found solution satisfies the original equation.

Solving an equation correctly requires understanding the type of equation you are dealing with and the right method to use. For cubic equations, applying the technique of extracting the cube root is an effective way to find the variable value.

Mathematical functions describe relationships between inputs and outputs. A function takes an input, processes it, and provides a single output. For an equation to define a function, every input must yield exactly one output.

In our exercise, \(y = \sqrt[3]{27 - x}\) is confirmed to be a function because each \(x\) value results in only one \(y\) value. This implies it's a legitimate function: each input \(x\) maps to one and only one output \(y\).
Mathematical functions are fundamental in various fields of study since they model and describe real-world phenomena succinctly. Functions can be classified into various types such as linear, quadratic, and cubic, depending on the nature of the equation that defines them.