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Chapter 5: Problem 21

Find the domain of each rational function. $$ R(x)=\frac{x}{x^{3}-64} $$

Short Answer

Expert verified
The domain is all real numbers except x = 4.

Step by step solution

01

Understand the Rational Function

A rational function is defined for all values of x except where the denominator is zero because division by zero is undefined. In this exercise, we need to find where the denominator, which is \( x^3 - 64 \), equals zero.
02

Set the Denominator Equal to Zero

Set the denominator equal to zero to find the values that are not in the domain of the function:\[ x^3 - 64 = 0 \]
03

Solve the Equation

Solve the equation to find the values of x:\[ x^3 = 64 \]Take the cube root of both sides:\[ x = \sqrt[3]{64} \]Since \sqrt[3]{64} = 4, we get:\[ x = 4 \]
04

Determine the Domain

The function is not defined when x = 4. Therefore, the domain of the rational function excludes this value. The domain is all real numbers except x = 4:\[ \text{Domain} = \{ x \in \mathbb{R}\ | \ x \/ = 4 \} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Function
A rational function is a fraction where both the numerator and the denominator are polynomials. The general form of a rational function looks like this: \( R(x) = \frac{P(x)}{Q(x)} \). Here, \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) cannot be zero because division by zero is undefined. Rational functions can have zeros, holes, and vertical or horizontal asymptotes in their graphs. These behaviors are closely related to their denominators and numerators.
Domain Exclusion
The domain of a function is the set of all possible input values (x-values) that will give a real output. For rational functions, the domain excludes values that make the denominator zero because it would cause the function to be undefined.
To find these excluded values, set the denominator equal to zero and solve for x. In the case of the given function \( R(x) = \frac{x}{x^{3}-64} \), set the denominator \( x^3 - 64 \) to zero and solve:
  • \( x^3 - 64 = 0 \)

  • \( x^3 = 64 \)

  • \( x = \sqrt[3]{64} = 4 \)
So, the function \ R(x) \ is undefined when \ x = 4 \. The domain is all real numbers except \ x = 4 \.
Solving Equations
Solving equations often involves isolating the variable on one side. In this exercise, solving the equation \[ x^3 - 64 = 0 \] helps us find which values are not part of the domain.
Here's how to solve it step-by-step:
  • Start with the equation: \[ x^3 - 64 = 0 \]

  • Add 64 to both sides: \[ x^3 = 64 \]

  • Take the cube root of both sides to isolate x: \[ x = \sqrt[3]{64} \]

  • Simplify: \ x = 4 \
This approach shows why \ x = 4 \ is excluded from the domain, as it makes the denominator zero. Knowing how to solve such equations is key to understanding the behavior of rational functions.

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