Problem 19 - College Algebra

Question

In Exercises \(9-20,\) determine whether each equation defines \(y\) as a function of \(x .\) $$x+y^{3}=8$$

Step-by-Step Solution

Answer

Yes, the equation \(x+y^3=8\) defines \(y\) as a function of \(x\).

Step 1: Rearrange the Equation for y

To see if \(y\) is a function of \(x\), we start by isolating \(y\) on one side of the equation. We can do this by subtracting \(x\) from both sides of the equation. This gives us the relation \(y^3 = 8-x\).

Step 2: Simplify Equation

Then, to further solve for \(y\), we take the cube root of both sides. This results into \(y = \sqrt[3]{8-x}\).

Step 3: Verify if y is a function of x

We have expression \(y = \sqrt[3]{8-x}\). Since any real value of \(x\) gives us a unique real value of \(y\), \(y\) is indeed a function of \(x\). One key property of a function is that each input (\(x\) value) corresponds to exactly one output (\(y\) value). Thus the given equation represents \(y\) as a function of \(x\).

Key Concepts

Defining Functions Algebraically

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.

Cube Root Function

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.

Unique Output of Function

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.