Problem 71 - College Algebra with Modeling and Visualization

Question

Determine if the following operation describes a function. Explain your answer. Calculating the cube root of a number.

Step-by-Step Solution

Answer

Yes, calculating the cube root describes a function because each input has exactly one output.

Step 1: Define What a Function Is

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In other words, for each element in the domain, there should be one and only one element in the range.

Step 2: Analyze the Operation

The operation given is calculating the cube root of a number. Mathematically, if we denote a number as \( x \), the cube root of \( x \) is written as \( \sqrt[3]{x} \).

Step 3: Evaluate Input and Output

For any real number \( x \), there exists exactly one real number \( y \) such that \( y^3 = x \). This means that, for each input \( x \), the operation produces exactly one cube root \( y \).

Step 4: Verify Function Criteria

Since for each real number \( x \), there is one and only one output \( y \) which is its cube root, the cube root operation satisfies the definition of a function.

Key Concepts

Cube Root Operation

In mathematics, a cube root operation is used to find a number that, when multiplied by itself three times (\( x \times x \times x \)), gives the original number. This is often represented as \( \sqrt[3]{x} \). The cube root operation is a specific example of a root operation which is the inverse of exponentiation.
This means that if you have a number \( x \), the cube root is the value \( y \) such that \( y^3 = x \). Unlike square roots, the cube root of a number uniquely exists for any real number, whether positive, negative, or zero.
If \( x \) is positive, \( \sqrt[3]{x} \) will be positive; if \( x \) is negative, \( \sqrt[3]{x} \) will also be negative, reflecting its nature of being naturally defined across the entire number line. The operation is important because it helps simplify complex equations and is frequently used in radical equations and solving for variables.

Real Numbers

The real numbers are a system of numbers that includes every possible value along the number line. This includes:

• All rational numbers, which can be expressed as fractions (like 1/2, 2/3),
• Irrational numbers, which cannot be precisely written as simple fractions (such as \( \pi \) or \( \sqrt{2} \)),
• Whole numbers and integers, which are either non-fractional positive numbers (such as 1, 2, 3...), zero, or negative integers (-1, -2, etc.).

Understanding real numbers is crucial when dealing with functions since many operations and equations operate within this number set.
When evaluating the cube root operation, it’s important to remember that any real number \( x \) will have a real cube root. This property assures us that when we take the cube root of any real number, we remain within the set of real numbers, maintaining consistency in mathematical operations.

Function Criteria

The cube root operation meets the criteria of a function in mathematics. The basic definition of a function is a relationship that maps each input to a single output. In this context:

• The domain refers to the set of all possible input values, which in the case of cube root includes all real numbers.
• The range refers to all possible outputs, which again consist of all real numbers, as every real number has a unique cube root.

When proving the function criteria for a cube root, we rely on the idea that each number \( x \) has precisely one corresponding \( y \) such that \( y^3 = x \).
This consistency guarantees that for any input from the domain, the operation yields a single output in the range. Hence, the cube root operation satisfies the requirements needed to be called a function, maintaining a one-to-one correspondence between inputs and outputs.