Find the domain and range of `f(x)=(x-3)/(4-x).`

Explanation

To find the domain and range of the function \( f(x) = \frac{x-3}{4-x} \), we will follow these steps:

### Step 1: Determine the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. Since \( f(x) \) is a rational function, it is defined as long as the denominator is not equal to zero.

1. Set the denominator equal to zero to find the restriction:
\[ 4 - x = 0 \]
Solving for \( x \):
\[ x = 4 \]

2. Therefore, the function is not defined at \( x = 4 \). The domain of \( f(x) \) is all real numbers except \( 4 \):
\[ \text{Domain} = \mathbb{R} \setminus \{4\} \]

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

1. Set \( f(x) = y \):
\[ y = \frac{x-3}{4-x} \]

2. Cross-multiply to eliminate the fraction:
\[ y(4 - x) = x - 3 \]
Expanding this gives:
\[ 4y - xy = x - 3 \]

3. Rearranging the equation to isolate \( x \):
\[ xy + x = 4y + 3 \]
Factor out \( x \):
\[ x(y + 1) = 4y + 3 \]
Thus, solving for \( x \):
\[ x = \frac{4y + 3}{y + 1} \]

4. The expression for \( x \) is valid as long as the denominator is not zero:
\[ y + 1 \neq 0 \Rightarrow y \neq -1 \]

5. Therefore, the range of \( f(x) \) is all real numbers except \( -1 \):
\[ \text{Range} = \mathbb{R} \setminus \{-1\} \]

### Final Answer
- **Domain**: \( \mathbb{R} \setminus \{4\} \)
- **Range**: \( \mathbb{R} \setminus \{-1\} \)