find the domain of f(x)=x over the square root of x squared minus 3x minus 4. write in interval notation 

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.

find the domain of f(x)=x over the square root of x squared minus 3x minus 4. write in interval notation

Learn how to find the domain of the function f(x) = x over the square root of x² - 3x - 4. This step-by-step guide covers solving quadratic equations, factoring, and using a sign chart to determine where the function is defined. Suitable for students in Grades 9-12.

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