Find the domain and range of real function ,`f(x)=|x+1|/(x+1)`

Answer

To find the domain and range of the function \( f(x) = \frac{|x + 1|}{x + 1} \), we will follow these steps:

### Step 1: Determine the Domain
The function \( f(x) \) is defined as long as the denominator is not zero. Therefore, we need to find when \( x + 1 = 0 \).

\[ x + 1 = 0 \implies x = -1 \]

Since \( f(x) \) becomes undefined when \( x = -1 \), the domain of \( f(x) \) excludes this value. Thus, the domain is:

\[ \text{Domain} = \{ x \in \mathbb{R} : x \neq -1 \} \]

### Step 2: Analyze the Function Based on the Absolute Value
Next, we will analyze the function based on the definition of the absolute value. The absolute value function \( |x + 1| \) can be expressed in two cases:

1. **Case 1:** When \( x + 1 \geq 0 \) (i.e., \( x \geq -1 \)):
\[ f(x) = \frac{x + 1}{x + 1} = 1 \quad \text{for } x > -1 \]

2. **Case 2:** When \( x + 1 < 0 \) (i.e., \( x < -1 \)):
\[ f(x) = \frac{-(x + 1)}{x + 1} = -1 \]

### Step 3: Determine the Range
From the analysis above, we find that:

- For \( x > -1 \), \( f(x) = 1 \).
- For \( x < -1 \), \( f(x) = -1 \).

Thus, the function takes on the values \( 1 \) and \( -1 \) depending on the interval of \( x \). Therefore, the range of \( f(x) \) is:

\[ \text{Range} = \{-1, 1\} \]

### Final Answer
- **Domain:** \( \{ x \in \mathbb{R} : x \neq -1 \} \)
- **Range:** \( \{-1, 1\} \)