Find the Domain and Range of \(f(x)=\frac{x-2}{2-x}\)
Question:
Find the domain and range of the real valued function:
$$
f(x)=\frac{x-2}{2-x}
$$
Solution
Domain
Given: $$ f(x)=\frac{x-2}{2-x} $$
The denominator cannot be zero.
Therefore, $$ 2-x\ne0 $$
$$ x\ne2 $$
Hence, the domain is: $$ \mathbb{R}-\{2\} $$
Range
Simplify the function: $$ f(x)=\frac{x-2}{-(x-2)} $$
$$ f(x)=-1 \quad (x\ne2) $$
Therefore, the function always gives the value \(-1\).
Hence, the range is: $$ \{-1\} $$