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M05 Homework 5.3
Question 3, 6.3.21

Find the domain of the following rational function:

[tex]\[ R(x) = \frac{x}{x^3 - 216} \][/tex]

Select the correct choice below and fill in any answer boxes within your choice.

A. The domain of [tex]\( R(x) \)[/tex] is [tex]\(\{ x \mid \square \} \)[/tex].
(Type an inequality in the form [tex]\( x \neq \)[/tex]. Use integers or fractions for any numbers in the expression. Use a comma to separate answers.)

B. The domain of [tex]\( R(x) \)[/tex] is the set of all real numbers.


Sagot :

To determine the domain of the rational function [tex]\( R(x) = \frac{x}{x^3 - 216} \)[/tex], we need to identify the values of [tex]\( x \)[/tex] that would make the denominator zero, as these values are where the function is undefined.

### Step-by-Step Solution

1. Identify the Denominator:
The denominator of [tex]\( R(x) \)[/tex] is [tex]\( x^3 - 216 \)[/tex].

2. Set the Denominator to Zero:
To find the values that make the denominator zero, we solve the equation:
[tex]\[ x^3 - 216 = 0 \][/tex]

3. Solve the Equation:
This can be solved by factoring or using the cubic root. Recognize that [tex]\( 216 = 6^3 \)[/tex], so:
[tex]\[ x^3 - 6^3 = 0 \][/tex]
Which gives us:
[tex]\[ x^3 = 216 \][/tex]
Taking the cubic root on both sides, we find:
[tex]\[ x = 6 \][/tex]

4. Identify Complex Roots:
In polynomial equations, there might be complex roots as well. Solving the polynomial [tex]\( x^3 - 216 \)[/tex] using factoring or specialized methods will give us:
[tex]\[ x = 6, \quad x = -3 - 3\sqrt{3}i, \quad x = -3 + 3\sqrt{3}i \][/tex]
This means the denominator is zero at these [tex]\( x \)[/tex] values: [tex]\( 6 \)[/tex], [tex]\( -3 - 3\sqrt{3}i \)[/tex], and [tex]\( -3 + 3\sqrt{3}i \)[/tex].

5. Determine the Domain:
The domain of [tex]\( R(x) \)[/tex] must exclude the [tex]\( x \)[/tex] values that make the denominator zero. Hence, [tex]\( x \)[/tex] cannot be:
[tex]\[ 6, \quad -3 - 3\sqrt{3}i, \quad -3 + 3\sqrt{3}i \][/tex]

So, the domain of [tex]\( R(x) \)[/tex] is all real numbers excluding [tex]\( x = 6 \)[/tex]. The complex roots do not affect the domain for real number inputs, so we only have to exclude the real number root.

### Correct Option:
The correct choice given in the problem statement is:
A. The domain of [tex]\( R(x) \)[/tex] is [tex]\( \{ x \mid x \neq 6 \} \)[/tex].

You fill in the blank as [tex]\( \boxed{6} \)[/tex]. Hence, the completed answer choice is:
[tex]\[ \text{A. The domain of } R(x) \text{ is } \{ x \mid x \neq 6 \}. \][/tex]