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Chapter 13: Problem 12

Find the domain. $$ f(x)=\frac{11 x}{x^{3}+27} $$

Short Answer

Expert verified
Question: Determine the domain of the function $$f(x) = \frac{11x}{x^3 + 27}$$. Answer: The domain of the function is $$D_{f(x)} = (-\infty, -3) \cup (-3, \infty)$$, meaning all real numbers except for x = -3.

Step by step solution

01

Identify the given function

We are given the function $$f(x) = \frac{11x}{x^3 + 27}$$ and need to find its domain.
02

Check if the denominator equals zero

In order to find the domain of the function, we must determine when the denominator equals zero. We do this by solving the following equation: $$x^3 + 27 = 0$$
03

Solve the equation

Solving for x, we get the following: $$x^3 = -27$$ $$x = \sqrt[3]{-27}$$ $$x = -3$$ So, the denominator equals zero when x = -3.
04

Determine the domain

Since the function is undefined when x = -3, the domain of the function is all real numbers except for x = -3. This can be written in interval notation as: $$D_{f(x)} = (-\infty, -3) \cup (-3, \infty)$$

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Key Concepts

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

Rational Functions
Rational functions are an important concept in mathematics. They are functions where a polynomial is divided by another polynomial. The general form of a rational function is given by \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
Rational functions are similar to fractions, but instead of numbers, they use expressions. This comes with additional conditions and properties.
  • The most important feature of rational functions is their domain. The domain consists of all real numbers except where the denominator is zero.
  • As with fractions, division by zero is undefined for rational functions.
This means solving rational functions often involves studying the denominator carefully to ensure it does not equal zero, and thus calculate the range of permissible x-values for the domain.
Denominator
The denominator in a rational function is crucial. It's the expression beneath the fraction line, and it significantly determines the function's properties and limitations.
In the given exercise, the denominator is \( x^3 + 27 \).
Why is it so important? Because the denominator must never be zero. A zero denominator would make the function undefined.
  • To find when the denominator is zero, set the denominator equal to zero and solve the resulting equation.
  • In the example, solving \( x^3 + 27 = 0 \) leads to \( x = -3 \).
This critical x-value, when substituted into the denominator, results in a zero division鈥攚hich is not allowed. Thus, we exclude \( x = -3 \) from the domain of the function.
Interval Notation
Interval notation is a way mathematicians communicate the domain of a function concisely and clearly. It utilizes brackets and parentheses to show inclusive and exclusive boundaries respectively.
In our example, the domain cannot include \( x = -3 \) because it makes the denominator zero. This makes the interval of all real numbers except \( -3 \).
  • The correct interval notation shows the domain as \( (-\infty, -3) \cup (-3, \infty) \).
Here's how this notation works:
* \((-\infty, -3)\) represents all numbers less than -3.
* \((-3, \infty)\) represents all numbers greater than -3.
The union symbol \( \cup \) is used to show that these two intervals together form the complete domain without -3.
Cubic Root
Understanding cubic roots is essential when dealing with polynomials that result in expressions like \( x^3 + 27 \). A cubic root is the number that, when multiplied by itself twice (used in three factors), gives the original number.
So, for \( x^3 = -27 \), the cubic root of \(-27\) is the solution.
  • The cubic root symbol \( \sqrt[3]{...} \) is used to denote this. In the example, \( x = \sqrt[3]{-27} \).
  • Solving further, \( x = -3 \) because \( (-3) \times (-3) \times (-3) = -27 \).
This straightforward calculation shows that \( -3 \) is the only real root for \( x \) in \( x^3 + 27 = 0 \). Cubic roots are especially unique since every real number has exactly one real cubic root, making them often simpler to handle than square roots.

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