Find the Domain and Range of the Function
Question:
The domain and range of the real function
\[ f(x)=\frac{4-x}{x-4} \]
are given by
(a) Domain \(=R\), Range \(=\{-1,1\}\)
(b) Domain \(=R-\{1\}\), Range \(=R\)
(c) Domain \(=R-\{4\}\), Range \(=\{-1\}\)
(d) Domain \(=R-\{-4\}\), Range \(=\{-1,1\}\)
Solution:
\[ f(x)=\frac{4-x}{x-4} \]
\[ =\frac{-(x-4)}{x-4} \]
\[ =-1 \qquad (x\ne4) \]
Therefore,
Domain:
\[ R-\{4\} \]
Range:
\[ \{-1\} \]
Hence,
\[ \boxed{\text{Correct Answer: (c)}} \]