Find the domain of the given function.
​F(x) = StartFraction x minus 9 Over x cubed plus 3 x EndFraction

To find the domain of the function \( F(x) = \frac{x - 9}{x^3 + 3x} \), we need to determine the values of \( x \) for which the function is defined. The function \( F(x) \) will be undefined wherever the denominator is equal to zero.

The denominator of \( F(x) \) is \( x^3 + 3x \). We need to find the values of \( x \) that make the denominator zero:

\[ x^3 + 3x = 0 \]

Factor the expression:

\[ x(x^2 + 3) = 0 \]

Setting each factor equal to zero gives us:

\[ x = 0 \]

\[ x^2 + 3 = 0 \]

Solving \( x^2 + 3 = 0 \):

\[ x^2 = -3 \]

\[ x = \pm \sqrt{-3} \]

Since \( \sqrt{-3} \) is not a real number, the only real solution is \( x = 0 \).

Thus, the function \( F(x) \) is undefined at \( x = 0 \).

Therefore, the domain of \( F(x) \) is all real numbers except \( x = 0 \). In interval notation, the domain is:

\[ (-\infty, 0) \cup (0, \infty) \]

Would you like more details or have any questions?

Here are some questions you might want to ask next:
1. How do you find the domain of a function involving square roots?
2. Can you explain the concept of asymptotes in rational functions?
3. How do you determine the range of a rational function?
4. What are the steps to simplify a rational expression?
5. How do you find the domain of a function involving logarithms?

**Tip:** When dealing with rational functions, always set the denominator equal to zero and solve for \( x \) to identify the values that are excluded from the domain.

Discover how to find the domain of the function F(x) = (x - 9) / (x^3 + 3x) step-by-step. Learn about rational functions and determining the domain based on the denominator being non-zero. Suitable for Grades 10-12.

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