The domain of the function f(x) = `(|x + 3|)/(x + 3) ` is

Options

A. {-3}
B. R - {-3}
C. R - {3}
D. R

Answer

To find the domain of the function \( f(x) = \frac{|x + 3|}{x + 3} \), we need to determine the values of \( x \) for which this function is defined.

### Step-by-Step Solution:

1. **Identify the Function**:
The function is given as \( f(x) = \frac{|x + 3|}{x + 3} \).

2. **Understand the Absolute Value**:
The absolute value \( |x + 3| \) is defined for all real numbers. However, we need to consider the denominator \( x + 3 \) as well.

3. **Set the Denominator Not Equal to Zero**:
For the function to be defined, the denominator cannot be zero. Therefore, we set:
\[ x + 3 \neq 0 \]

4. **Solve for x**:
Solving the inequality gives:
\[ x \neq -3 \]

5. **Determine the Domain**:
The domain of the function is all real numbers except \( -3 \). In set notation, we can express this as:
\[ \text{Domain} = \{ x \in \mathbb{R} : x \neq -3 \} \]

6. **Conclusion**:
Thus, the domain of the function \( f(x) = \frac{|x + 3|}{x + 3} \) is all real numbers except \( -3 \).

### Final Answer:
The domain of the function is \( \mathbb{R} \setminus \{-3\} \) or \( (-\infty, -3) \cup (-3, \infty) \).

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