If `f(x)=(1)/(sqrt(x))`, what is the domain of f(x)?

Options

A. All real numbers
B. All integers except zero
C. All positive real numbers
D. All real numbers except zero

Explanation

To find the domain of the function \( f(x) = \frac{1}{\sqrt{x}} \), we need to determine the set of values for \( x \) that will make the function defined. Here are the steps to solve the problem:

### Step 1: Identify the conditions for the square root
The expression under the square root, \( x \), must be non-negative because the square root of a negative number is not defined in the set of real numbers. Therefore, we have:
\[ x \geq 0 \]

### Step 2: Exclude values that make the function undefined
Since the function is defined as \( f(x) = \frac{1}{\sqrt{x}} \), we need to ensure that the denominator is not zero. The square root of zero is zero, and division by zero is undefined. Thus, we need to exclude \( x = 0 \):
\[ x > 0 \]

### Step 3: Combine the conditions
From the above steps, we conclude that \( x \) must be greater than zero. Therefore, the domain of \( f(x) \) is:
\[ \text{Domain of } f(x) = (0, \infty) \]

### Final Answer
The domain of \( f(x) = \frac{1}{\sqrt{x}} \) is all positive real numbers, which can be expressed in interval notation as \( (0, \infty) \).

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