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Math  /  Algebra

Question31) xx−4\frac{x}{\sqrt{x-4}} A) {x∣x≥4}\{x \mid x \geq 4\} B) all real numbers C) {x∣x≠4}\{x \mid x \neq 4\} D) {x∣x>4}\{x \mid x>4\}

Studdy Solution

STEP 1

1. The expression xx−4\frac{x}{\sqrt{x-4}} involves a square root in the denominator.
2. The denominator must be non-zero for the expression to be defined.
3. The radicand (inside the square root) must be non-negative for the square root to be real.

STEP 2

1. Identify the condition for the square root to be defined.
2. Identify the condition for the denominator to be non-zero.
3. Combine the conditions to determine the domain.
4. Match the domain with the given options.

STEP 3

Identify the condition for the square root to be defined:
The expression inside the square root is x−4x - 4. For the square root to be defined, we need:
x−4≥0 x - 4 \geq 0
This simplifies to:
x≥4 x \geq 4

STEP 4

Identify the condition for the denominator to be non-zero:
Since the square root is in the denominator, it cannot be zero. Therefore, we need:
x−4≠0 \sqrt{x - 4} \neq 0
This implies:
x−4≠0 x - 4 \neq 0
Which simplifies to:
x≠4 x \neq 4

STEP 5

Combine the conditions to determine the domain:
From Step 1, we have x≥4 x \geq 4 . From Step 2, we have x≠4 x \neq 4 .
Combining these, we get:
x>4 x > 4

STEP 6

Match the domain with the given options:
The domain we found is x>4 x > 4 , which corresponds to option D.
The correct answer is D\boxed{D}.

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