Find the Domain of \(f(x)=\frac{x^2+2x+1}{x^2-8x+12}\)
Question:
Find the domain of the following real valued function of real variable:
$$
f(x)=\frac{x^2+2x+1}{x^2-8x+12}
$$
Solution
Given: $$ f(x)=\frac{x^2+2x+1}{x^2-8x+12} $$
The denominator cannot be zero.
$$ x^2-8x+12\ne0 $$
Factorizing: $$ (x-6)(x-2)\ne0 $$
Therefore, $$ x\ne6 \quad \text{and} \quad x\ne2 $$
Hence, all real numbers except \(2\) and \(6\) are allowed.
Therefore, the domain is: $$ \mathbb{R}-\{2,6\} $$
or $$ (-\infty,2)\cup(2,6)\cup(6,\infty) $$