Find the Domain and Range of \(f(x)=\dfrac{2-x}{x-2}\)
Question
The domain and range of the function
\[ f(x)=\frac{2-x}{x-2} \]are __________ and __________ respectively.
Solution
Given
\[ f(x)=\frac{2-x}{x-2} \]Rewrite the numerator:
\[ 2-x=-(x-2) \]Therefore,
\[ f(x)=\frac{-(x-2)}{x-2} \] \[ f(x)=-1 \qquad \text{for } x\ne2 \]Domain
The denominator cannot be zero.
\[ x-2\ne0 \] \[ x\ne2 \]Hence the domain is
\[ \mathbb{R}\setminus\{2\} \]Range
Since
\[ f(x)=-1 \]for all values in the domain, the function takes only one value.
Hence the range is
\[ \{-1\} \]Final Answer
\[ \boxed{\mathbb{R}\setminus\{2\}} \]and
\[ \boxed{\{-1\}} \]