Question

Find the domain of the function using interval notation. F(x)=√x-6/√x-4​

Answer 1

Answer:

$(-\infty,4)\cup(6,+\infty)$

Step-by-step explanation:

The function is:

$f(x)=\frac{\sqrt{x-6}}{\sqrt{x-4}}=\sqrt{\frac{x-6}{x-4}}$

The rational function into the square root must be positive.

To find the domain you focus in two cases. One of theme, with both denominator and numerator are positive, that is:

$x-6>0\\\\x>6\\\\x-4>0\\\\x>4$

the intersection of these intervals is:

$(6,+\infty)\cap (4,+\infty)=(6,+\infty)$

In the other case, both, denominator and numerator are negative, that is:

$x-6<0\\\\x<6\\\\x-4<0\\\\x<4$

the intersection of these intervals is:

$(-\infty,6)\cap (-\infty,4)=(-\infty,4)$

Finally, the domain is the union:

$(-\infty,4)\cup(6,+\infty)$