Question

Find the domain of definition of the following function:
$y=\log \frac{x^2+8x+7}{x^2+7}.$

Answer 1

$y=log\left(\frac{x^2+8x+7}{x^2+7}\right)$

For $y$ to be real and defined.

$\frac{x^2+8x+7}{x^2+7}>0$

$x^2+7$ is always $>0$ and never be equal to zero.

Therefore, $x^2+8x+7>0\Rightarrow (x+1)(x+7)>0$

Therefore, $x\in (-\infty ,-7)\cup (-1,\infty )$

Therefore, domain of $y$ is $x\in (-\infty ,-7)\cup (-1,\infty ).$