Find the Domain of \(f(x)=\frac{x^2+2x+1}{x^2-8x+12}\)

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 $$ (-\infty,2)\cup(2,6)\cup(6,\infty) $$