The domain and range of the real function f defined by `f(x)=(4-x)/(x-4)` is (a) Domain =R , Range ={-1,2} (b) Domain =R -{1}, Range R (c) Domain =R -{4}, Range ={-1} (d) Domain =R -{-4}, Range ={-1,1}

Options

A. Domain =R , Range ={-1,2}
B. Domain =R -{1}, Range R
C. Domain =R -{4}, Range ={-1}
D. Domain =R -{-4}, Range ={-1,1}

Explanation

To solve the problem, we need to determine the domain and range of the function \( f(x) = \frac{4 - x}{x - 4} \).

### Step 1: Determine the Domain
The domain of a function consists of all the values of \( x \) for which the function is defined.

1. Identify the denominator: The denominator of the function is \( x - 4 \).
2. Set the denominator to zero to find the restriction:
\[ x - 4 = 0 \implies x = 4 \]
3. Since the function is undefined when \( x = 4 \), the domain of \( f(x) \) is all real numbers except 4.

Thus, the domain is:
\[ \text{Domain} = \mathbb{R} - \{4\} \]

### Step 2: Determine the Range
Next, we will find the range of the function.

1. Rewrite the function:
\[ f(x) = \frac{4 - x}{x - 4} = \frac{-(x - 4)}{x - 4} \]
2. Simplify the function:
\[ f(x) = -1 \quad \text{for } x \neq 4 \]
3. Since \( f(x) \) equals \(-1\) for all \( x \) in the domain, the range of the function is simply the single value \(-1\).

Thus, the range is:
\[ \text{Range} = \{-1\} \]

### Conclusion
Combining the results from the domain and range, we have:
- Domain: \( \mathbb{R} - \{4\} \)
- Range: \( \{-1\} \)

The correct option from the given choices is:
(c) Domain = \( \mathbb{R} - \{4\} \), Range = \{-1\}