Find the domains and ranges of the functions.
`(x-1)/(x+2)`

Answer

To find the domain and range of the function \( f(x) = \frac{x-1}{x+2} \), we will follow these steps:

### Step 1: Determine the Domain
The domain of a function consists of all the values of \( x \) for which the function is defined. The function \( f(x) \) is undefined when the denominator is equal to zero.

Set the denominator equal to zero:
\[ x + 2 = 0 \]
Solving for \( x \):
\[ x = -2 \]

Thus, the function is undefined at \( x = -2 \). Therefore, the domain of the function is all real numbers except \( -2 \):
\[ \text{Domain} = \mathbb{R} \setminus \{-2\} \]

### Step 2: Determine the Range
To find the range, we need to express \( y \) in terms of \( x \) and then determine the values that \( y \) can take.

Let:
\[ y = \frac{x-1}{x+2} \]

Rearranging this equation to solve for \( x \):
\[ y(x + 2) = x - 1 \]
Expanding and rearranging gives:
\[ yx + 2y = x - 1 \]
\[ yx - x = -1 - 2y \]
Factoring out \( x \):
\[ x(y - 1) = -1 - 2y \]
Now, solving for \( x \):
\[ x = \frac{-1 - 2y}{y - 1} \]

Next, we need to find when this expression is defined. The expression for \( x \) is undefined when the denominator is zero:
\[ y - 1 = 0 \]
Thus:
\[ y \neq 1 \]

This means that the function can take all real values except \( 1 \). Therefore, the range of the function is:
\[ \text{Range} = \mathbb{R} \setminus \{1\} \]

### Final Answer
- **Domain**: \( \mathbb{R} \setminus \{-2\} \)
- **Range**: \( \mathbb{R} \setminus \{1\} \)