91Ó°ÊÓ

When we talk about defining functions algebraically, we refer to the process of expressing a relationship between two variables in a precise, equation-based format. In the context of the given exercise, we're interested in establishing whether the equation presented indeed describes the variable y as a function of variable x.

To do this, we arrange the equation to isolate y on one side, thus making it clear how y is determined by x. By performing algebraic manipulations, such as subtracting x from both sides and then taking the cube root (the focus of the next section), we reformulate the initial equation into a form where for each x, there is a specific y outcome. This transformation reflects the essence of a function: a single output for every distinct input.

The cube root function, symbolized as \( \sqrt[3]{\text{\textbullet}} \), is an essential concept when dealing with equations where the variable is raised to the power of three. By applying the cube root to both sides of an equation, as seen in the solution, we effectively reverse the process of cubing a number.

Using the given exercise as an example, after algebraic manipulation, we arrive at the equation \( y^3 = 8 - x \). To find y, we apply the cube root function to both sides, resulting in \( y = \sqrt[3]{8 - x} \). This function is particularly noteworthy because it always provides a real number as an output for any real number input. This characteristic ensures that it translates to a single, unique value of y for each value of x—a fundamental property for a function in mathematics.

Ensuring that each input has a unique output is the cornerstone of the definition of a function in mathematics. When we assign a value to x, the rules of the function dictate that only one value can correspond to y. This concept guarantees consistency and predictability within mathematical relationships and models.

In this exercise, the equation \( y = \sqrt[3]{8 - x} \) demonstrates that no matter what real value of x we substitute into the equation, the cube root operation assures us of obtaining a unique value for y. This one-to-one relationship between x and y qualifies the equation as a function. It is also important to note that every valid x leads to a valid y, highlighting the function's domain and range, which in this case, are all real numbers.