if f(x) = `sqrt( x-1)/(x-4)` defines a function of R. then what is domain?

Options

A. `(-infty,4)u(4,infty)`
B. `(4,infty)`
C. `(1,4)u(4,infty)`
D. `1,4u(4,infty)`

Answer

To find the domain of the function \( f(x) = \frac{\sqrt{x - 1}}{x - 4} \), we need to consider two conditions:

1. The expression inside the square root must be non-negative.
2. The denominator must not be zero.

### Step 1: Determine the condition for the square root
The expression inside the square root is \( x - 1 \). For the square root to be defined, we need:
\[ x - 1 \geq 0 \]
This simplifies to:
\[ x \geq 1 \]

### Step 2: Determine the condition for the denominator
The denominator is \( x - 4 \). For the function to be defined, the denominator must not be zero:
\[ x - 4 \neq 0 \]
This simplifies to:
\[ x \neq 4 \]

### Step 3: Combine the conditions
From Step 1, we have \( x \geq 1 \). From Step 2, we have \( x \neq 4 \).

Now we need to express the domain considering both conditions:
- The values of \( x \) must be greater than or equal to 1.
- However, \( x \) cannot be equal to 4.

Thus, we can express the domain in interval notation as:
\[ [1, 4) \cup (4, \infty) \]

### Final Answer
The domain of the function \( f(x) = \frac{\sqrt{x - 1}}{x - 4} \) is:
\[ [1, 4) \cup (4, \infty) \]

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