Find the domain of `f(x)=(1)/(sqrt(5-x)`

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{5 - x}} \), we need to ensure that the expression under the square root is positive and not equal to zero, as the square root of a negative number is not defined in the real number system, and division by zero is undefined.

### Step 1: Set up the inequality
We start with the condition that the expression inside the square root must be greater than zero:
\[ 5 - x > 0 \]

### Step 2: Solve the inequality
To solve this inequality, we can rearrange it:
\[ 5 > x \]
or equivalently,
\[ x < 5 \]

### Step 3: Check for equality
Next, we need to ensure that the expression does not equal zero:
\[ 5 - x \neq 0 \]
This implies:
\[ x \neq 5 \]

### Step 4: Combine the conditions
From the above steps, we have:
1. \( x < 5 \)
2. \( x \neq 5 \)

Since \( x < 5 \) already excludes \( x = 5 \), we can conclude that the domain of \( f(x) \) is all real numbers less than 5.

### Step 5: Write the domain in interval notation
Thus, the domain of the function can be expressed in interval notation as:
\[ (-\infty, 5) \]

### Final Answer
The domain of \( f(x) = \frac{1}{\sqrt{5 - x}} \) is \( (-\infty, 5) \).

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