To determine the domain of the function:

$f(x) = \frac{9x + 10}{x^2 - 3}$

we need to find the values of $x$ for which the function is defined. A rational function is undefined where the denominator equals zero.

Step 1: Set the denominator equal to zero

$x^2 - 3 = 0$

Step 2: Solve for $x$

$x^2 = 3$

$x = \pm\sqrt{3}$

Step 3: Determine the domain

Since the function is undefined at $x = \pm\sqrt{3}$, the domain consists of all real numbers except these points.

Thus, the domain is:

$\mathbb{R} \setminus \{\pm\sqrt{3}\}$

Or in interval notation:

$(-\infty, -\sqrt{3}) \cup (-\sqrt{3}, \sqrt{3}) \cup (\sqrt{3}, \infty)$

Would you like a graphical representation of this function for better visualization?

Related Questions:

1. How do you determine the vertical asymptotes of a rational function?
2. What is the range of the function $f(x) = \frac{9x+10}{x^2-3}$?
3. How do you find the horizontal asymptote of a rational function?
4. What happens to the function as $x \to \infty$ or $x \to -\infty$?
5. Can a rational function have a hole in its graph?

Tip:

For any rational function, the domain is all real numbers except where the denominator is zero.