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

Options

A. `(-oo,4)uu(4,oo)`
B. `[4,oo]`
C. `(1,4)uu(4,oo)`
D. `[1,4)uu(4,oo)`

Explanation

To find the domain of the function \( f(x) = \frac{\sqrt{x-1}}{x-4} \), we need to determine the values of \( x \) for which the function is defined.

### Step-by-Step Solution:

1. **Identify the Square Root Condition**:
The expression under the square root, \( x - 1 \), must be non-negative for the square root to be defined. Therefore, we set up the inequality:
\[ x - 1 \geq 0 \]
Solving this gives:
\[ x \geq 1 \]

2. **Identify the Denominator Condition**:
The denominator \( x - 4 \) cannot be zero, as division by zero is undefined. Therefore, we set up the equation:
\[ x - 4 \neq 0 \]
Solving this gives:
\[ x \neq 4 \]

3. **Combine the Conditions**:
From the first condition, we have \( x \geq 1 \). From the second condition, we have \( x \neq 4 \). Thus, the values of \( x \) that satisfy both conditions are:
\[ x \in [1, 4) \cup (4, \infty) \]

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