Find the domain and range of real function ,`f(x)=(4-x)/(x-4)`

Answer

To find the domain and range of the function \( f(x) = \frac{4 - x}{x - 4} \), 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.

1. **Identify the denominator**: The function is undefined when the denominator is zero.
\[ x - 4 = 0 \]
Solving this gives:
\[ x = 4 \]

2. **State the domain**: Therefore, the function is defined for all real numbers except \( x = 4 \). We can express the domain in interval notation:
\[ \text{Domain} = (-\infty, 4) \cup (4, \infty) \]

### Step 2: Determine the Range
To find the range, we need to analyze the output values (y-values) of the function.

1. **Set the function equal to y**:
\[ y = \frac{4 - x}{x - 4} \]

2. **Rearrange the equation**: We can manipulate the equation to express it in a different form:
\[ y(x - 4) = 4 - x \]
Expanding gives:
\[ yx - 4y = 4 - x \]
Rearranging this yields:
\[ yx + x = 4 + 4y \]
Factoring out x:
\[ x(y + 1) = 4 + 4y \]
Thus:
\[ x = \frac{4 + 4y}{y + 1} \]

3. **Identify restrictions on y**: The expression for \( x \) is valid as long as \( y + 1 \neq 0 \) (to avoid division by zero). This gives:
\[ y \neq -1 \]

4. **State the range**: Therefore, the range of the function is all real numbers except \( y = -1 \). We can express the range in interval notation:
\[ \text{Range} = (-\infty, -1) \cup (-1, \infty) \]

### Final Answer
- **Domain**: \( (-\infty, 4) \cup (4, \infty) \)
- **Range**: \( (-\infty, -1) \cup (-1, \infty) \)