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Given the functions [tex]\( f(x) = x^3 + 2 \)[/tex] and [tex]\( g(x) = x - 9 \)[/tex], perform the indicated operation and state the domain restriction.

[tex]\[
\left(\frac{f}{g}\right)(x)
\][/tex]

A. [tex]\(\frac{x^3 + 2}{x - 9}, x \neq 9\)[/tex]

B. [tex]\(\frac{x - 9}{x^3 + 2}, x \neq 9\)[/tex]

C. [tex]\(\frac{x^3 + 2}{x - 9}, x \neq -9\)[/tex]

D. [tex]\(\frac{x - 9}{x^3 + 2}, x \neq -9\)[/tex]


Sagot :

Let's take this step by step to solve the given problem.

First, recall the functions [tex]\( f(x) = x^3 + 2 \)[/tex] and [tex]\( g(x) = x - 9 \)[/tex].

We are asked to perform the operation [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex], which means we need to find:

[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^3 + 2}{x - 9} \][/tex]

The domain restriction for this function occurs where the denominator is zero because division by zero is undefined. Therefore, we need to find the value of [tex]\( x \)[/tex] that makes the denominator [tex]\( g(x) = 0 \)[/tex]:

[tex]\[ x - 9 = 0 \implies x = 9 \][/tex]

Thus, we have the domain restriction [tex]\( x \neq 9 \)[/tex].

Now we can analyze the given options:

1. [tex]\(\frac{x^3 + 2}{x - 9}, x \neq -9\)[/tex]
2. [tex]\(\frac{x - 9}{x^3 + 2}, x \neq 9\)[/tex]
3. [tex]\(\frac{x^3 + 2}{x - 9}, x \neq 9\)[/tex]
4. [tex]\(\frac{x - 9}{x^3 + 2}, x \neq -9\)[/tex]

We already found that the correct form for [tex]\( \left(\frac{f}{g}\right)(x) \)[/tex] is:
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{x^3 + 2}{x - 9} \][/tex]
with the domain restriction [tex]\( x \neq 9 \)[/tex].

Now, let's evaluate the results of different options with specific input values for clarity:

Option 1: [tex]\(\left(\frac{f}{g}\right)(-9)\)[/tex]
[tex]\[ \left(\frac{-9^3 + 2}{-9 - 9}\right) = \frac{-729 + 2}{-18} = \frac{-727}{-18} = 40.388888888888886 \][/tex]

Option 2: [tex]\(\left(\frac{g}{f}\right)(9)\)[/tex]
[tex]\[ \left( \frac{9 - 9}{9^3 + 2}\right) = \frac{0}{729 + 2} = 0.0 \][/tex]

Option 3: [tex]\(\left(\frac{f}{g}\right)(9)\)[/tex]
[tex]\[ \frac{9^3 + 2}{9 - 9} = \frac{729 + 2}{0} \quad \text{undefined, as division by zero is not possible} \][/tex]

Option 4: [tex]\(\left(\frac{g}{f}\right)(-9)\)[/tex]
[tex]\[ \frac{-9 - 9}{(-9)^3 + 2} = \frac{-18}{-727} = 0.024759284731774415 \][/tex]

Given the correct domain restriction and the correct computed forms, the accurate options should be [tex]\(\frac{x^3 + 2}{x - 9}, x \neq 9\)[/tex].

Therefore, the correct answer is:

[tex]\[ \boxed{\frac{x^3 + 2}{x - 9}, x \neq 9} \][/tex]