Problem 20 - College Algebra

Question

In Exercises \(9-20,\) determine whether each equation defines \(y\) as a function of \(x .\) $$x+y^{3}=27$$

Step-by-Step Solution

Answer

Yes, the equation \(x + y^3 = 27\) does define \(y\) as a function of \(x\).

Step 1: Rearrange The Equation

Start by rearranging the equation to represent \(y\) as a function of \(x\). This can be done by subtracting \(x\) from both sides of the equation, resulting in \(y^3 = 27 - x\).

Step 2: Derive Function of y

To derive the function of \(y\), take the cube root of both sides of the equal sign. This will express \(y\) as a function of \(x\), i.e., \(y = \sqrt[3]{27 - x}\).

Step 3: Verify Whether y Is a Function of x

For \(y\) to be a function of \(x\), for each value of \(x\) there should be exactly one value of \(y\). Since the cube root will always yield a unique result for any real number, the given equation does indeed define \(y\) as a function of \(x\).

Key Concepts

Determining Functions

In algebra, determining whether an equation defines a function is crucial. A function from a set of inputs (usually called domain) to a set of outputs (usually called range) assigns each input exactly one output.

Here's how you can check if an equation provides a function:

• Start by expressing the equation explicitly, where one variable is solely dependent on another.
• Verify whether each input (or value of \(x\)) corresponds to just one output (or value of \(y\)).

Doing so ensures that the equation satisfies the definition of a function. In the exercise, rearranging the equation to form \(y = \sqrt[3]{27 - x}\) helps make this determination clear.

The cube root function directly links each \(x\) value to a singular \(y\) outcome, confirming it as a function.

Implicit Equations

Equations not explicitly solved for one variable are known as implicit equations. Understanding these is important because many real-world relationships are expressed implicitly.

How do you handle implicit equations?

• Reorganize the terms to isolate the desired variable.
• Transform the equation from implicit to explicit where applicable.

For the given exercise, the original form \(x + y^{3} = 27\) implicitly defines \(y\) in terms of \(x\). By rearranging to \(y^3 = 27 - x\), we turn it into an explicit equation, making it easier to study the function behavior.

This simplification turns challenging problems into more approachable tasks.

Cube Root Functions

Cube root functions are a type of radical function involving the cube root of a variable. Unlike square roots, cube roots are defined for all real numbers. This property plays a key role in algebra and functions.

Key points about cube root functions include:

• They map negative numbers as well as positives, unlike square roots.
• They offer a one-to-one correspondence between inputs and outputs, making them ideal for defining functions.

Applying this to the exercise, we find \(y = \sqrt[3]{27 - x}\), a cube root function, establishes a clear and unique relationship between \(x\) and \(y\).

The cube root's continuous and unambiguous nature guarantees that it behaves reliably across its domain.