Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(y^{3}=x^{2}\)

Step-by-Step Solution

Answer

Yes, \(y^3 = x^2\) represents \(y\) as a function of \(x\).

Step 1: Understanding Functions

A function is defined as a relation where each input is associated with exactly one output. Here, we need to determine if for every value of \(x\) there is a unique \(y\) value.

Step 2: Analyze the Equation

Consider the given equation \(y^3 = x^2\). We will analyze this to see if for every \(x\), there is exactly one \(y\).

Step 3: Solve for y

Solving the equation for \(y\), we have \(y = \sqrt[3]{x^2}\). This implies that the output \(y\) will be a real number cube root of \(x^2\).

Step 4: Determine Uniqueness of y

For each value of \(x\), \(x^2\) is non-negative, and thus has a unique real cube root \(y\). Since no \(x\) maps to more than one \(y\), \(y^3 = x^2\) represents \(y\) as a function of \(x\).

Key Concepts

Relation as a Function

A relation is simply a set of ordered pairs, usually representing the connection between two variables, such as \(x\) and \(y\). However, not every relation is a function. In mathematics, a function is a special type of relation. For a relation to qualify as a function, each \(x\) value, or input, must correspond to exactly one \(y\) value, or output. This fundamental characteristic forms the foundation of function definitions. It ensures that for every input in the domain of the function, there is a unique result in the range, helping in establishing predictable and consistent outcomes.

Understanding Unique Output in Functions

In context of a function, the principle of unique output refers to the necessity that every \(x\) value maps to only one \(y\) value. This means that no \(x\) should lead to multiple \(y\) outcomes. If it does, the relation fails to qualify as a function. In the example equation \(y^3 = x^2\), when solved for \(y\), we find \(y = \sqrt[3]{x^2}\). This solution indicates that for any real number \(x\), the resulting \(y\) value is unambiguously derived from the cube root of \(x^2\), ensuring one unique result. Unique outputs are crucial in functions for establishing reliable and consistent relationships between variables.

Input-Output Analysis

Input-output analysis in functions explores how different inputs generate outputs. It's essentially about playing with values to see how a relation operates. In \(y^3 = x^2\), understanding what happens to \(y\) when various \(x\) values are substituted helps us decode the behavior of the function.

• Input: Since \(x^2\) is non-negative for all real \(x\), the range of \(y\) is defined clearly.
• Output: The cube root operation \(\sqrt[3]{x^2}\) ensures that every input \(x\) translates to exactly one output \(y\), upholding the function property.

By analyzing the input-output relationship, we recognize the steadfast production of a single output value for any input value, solidifying the relationship as a function.

Solving Equations to Determine Functionality

When tasked with determining if a relation is a function, solving the equation is often a necessary step. Start by expressing the equation in a form where dependence is clear. In our case, starting with \(y^3 = x^2\), we solve for \(y\) to make the dependency explicit: \(y = \sqrt[3]{x^2}\). This step assures us that for each value of \(x\), \(x^2\) is unequivocally positive or zero, and thus has a unique cube root. Solving equations in this manner allows one to test the function criterion—the existence of a unique output for every input—and ensures clarity in whether a relation operates as a function.