We are asked to find the domain of the function:

$f(x) = \frac{x}{\sqrt{x^2 - 3x - 4}}$

To find the domain of the function, we need to determine where the function is defined. There are two things to consider:

1. The denominator cannot be zero.
2. The square root must have a non-negative argument.

Step 1: Solve for when the denominator is zero

We need to solve the equation $x^2 - 3x - 4 = 0$.

This is a quadratic equation, and we can solve it using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = -3$, and $c = -4$.

$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-4)}}{2(1)} = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm \sqrt{25}}{2}$

$x = \frac{3 \pm 5}{2}$

Thus, the two solutions are:

$x = \frac{3 + 5}{2} = 4 \quad \text{and} \quad x = \frac{3 - 5}{2} = -1$

So, $x = 4$ and $x = -1$ make the denominator zero, which means these values must be excluded from the domain.

Step 2: Ensure the argument inside the square root is non-negative

We now need to solve the inequality:

$x^2 - 3x - 4 \geq 0$

We already factored this quadratic equation:

$x^2 - 3x - 4 = (x - 4)(x + 1)$

Now solve the inequality:

$(x - 4)(x + 1) \geq 0$

We use a sign chart to determine where the product is non-negative. The critical points are $x = -1$ and $x = 4$.

1. For $x < -1$, both $x - 4$ and $x + 1$ are negative, so the product is positive.
2. For $-1 < x < 4$, one factor is negative and the other is positive, so the product is negative.
3. For $x > 4$, both factors are positive, so the product is positive.

Thus, the solution to the inequality is:

$x \in (-\infty, -1] \cup [4, \infty)$

Step 3: Combine the results

The domain of $f(x)$ is the set of all $x$ for which the denominator is not zero and the square root argument is non-negative. Since $x = -1$ and $x = 4$ make the denominator zero, we must exclude them from the domain.

Thus, the domain is:

$x \in (-\infty, -1) \cup (4, \infty)$

Final Answer:

The domain of $f(x)$ in interval notation is:

$(-\infty, -1) \cup (4, \infty)$

Would you like more details or have any questions?

---

Here are 5 related questions to deepen your understanding:

1. How do you determine the domain of a function with a square root in the denominator?
2. What happens when the denominator of a function equals zero?
3. Can a function have values where the square root of a negative number is involved?
4. How do you interpret the signs of factors in an inequality?
5. What is the significance of excluding certain values from the domain?

Tip: Always check both the numerator and denominator when finding the domain of a rational function.