Problem 21 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Concept of a Function

A relation represents as a function of x if for every input x, there is exactly one output . We need to verify if this condition is met for the given equation.

Step 2: Analyze the Given Equation

The equation provided is \( y = x^3 \). For each value of \( x \), we calculate \( x^3 \). This operation is a basic power function where for any given input \( x \), there is a single corresponding output \( y \).

Step 3: Evaluate the Uniqueness of Outputs

Since \( x \) raised to any power returns exactly one value for each specific \( x \), the output \( y \) is unique for every input \( x \).

Step 4: Conclusion on Functionality

Since each input \( x \) produces a unique output \( y \), the equation \( y = x^3 \) does represent \( y \) as a function of \( x \).

Key Concepts

Relations and Functions

In algebra, understanding the difference between relations and functions is crucial. A **relation** is a set of ordered pairs, usually represented as \( (x, y) \). In simple terms, any pairing of elements creates a relation. However, not all relations are functions. A **function** is a special type of relation.
What makes a function unique is its relationship between the elements: for every input value \( x \), there must be exactly one output value \( y \). This means:

• If you input a certain number into a function, you always get the same result for that number.
• In mappings or tables, each \( x \) value connects to only one \( y \) value.

By analyzing the concept of a function, we can determine it has a more constrained structure than a general relation. Hence, while all functions are relations, not every relation qualifies as a function.

Power Functions

Power functions play an essential role in algebra, specifically when analyzing different types of functions. A **power function** generally takes the form of \( y = x^n \), where \( n \) is any real number. Depending on the value of \( n \):

• When \( n \) is positive, the function grows or shrinks directly with \( x \).
• If \( n \) is negative or a fraction, the behavior of the function changes, potentially creating hyperbolic or root functions.

For the equation \( y = x^3 \), the power function specifically takes a cubic form. It implies for every \( x \), you multiply \( x \) by itself three times. This operation results always in one clear output value for each input \( x \). Power functions are foundational in calculus, as they demonstrate how variables can be manipulated by different powers.

Uniqueness of Outputs

In confirming whether an equation like \( y = x^3 \) functions appropriately, testing the **uniqueness of outputs** is vital. This principle ensures no ambiguity in results. For any given input \( x \), regardless of its value, the equation should provide exactly one output \( y \).
To understand the uniqueness:

• For a specific \( x \), plug it into the equation.
• If you always retrieve one output and no variations, the function is valid.

In the equation \( y = x^3 \), the uniqueness is evident because:

• Raising any real number \( x \) to the power of 3 consistently produces a distinct result.
• No matter if \( x \) is negative, positive, or zero, the output remains singular and definite.

This trait of uniqueness is what makes mathematical functions predictable and useful in computation and broader applications.