Question

Find the domain of the real valued function of real variable:  

(v)  \[f\left( x \right) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}\]

 

Answer 1

(v) Given:

\[f\left( x \right) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}\]

\[= \frac{x^2 + 2x + 1}{x^2 - 6x - 2x + 12}\]

\[= \frac{x^2 + 2x + 1}{x\left( x - 6 \right) - 2\left( x - 6 \right)}\]

\[= \frac{x^2 + 2x + 1}{\left( x - 6 \right)\left( x - 2 \right)}\]

Domain of f : Clearly,  f (x) is a rational function of x as 

\[\frac{x^2 + 2x + 1}{x^2 - 8x + 12}\] is a rational expression.
Clearly, f (x) assumes real values for all x except for all those values of x for which x² - 8x + 12 = 0, i.e. x = 2, 6. 
Hence, domain ( f ) = R -  {2,6}.