Question

Please give an explanation.

Answer 1

Answer:

option D

Step-by-step explanation:

So we know that $\sqrt{a} \sqrt{b} = \sqrt{ab}$

Applying this to our function, we have that:

$\sqrt{(x-5)} \sqrt{(x+2)} = \sqrt{(x+2)(x-5)}$

We know that the argument of a square root should always BE POSITIVE.

So we need to evaluate in which points the expression (x+2)(x-5) is positive.

So we know that (x+2) is possitive when x>-2 and negative when x<-2.

Also we know that (x-5) is possitive when x>5, if x<5 then x is negative.

Then we have:

if x<-2, then:

(x-5) is negative

(x+2) is negative

Then (x+2)(x-5) is positive.

If x>5 then:

(x-5) is positive

(x+2) is positive.

Then (x+2)(x-5) is positive.

If -2<x<5 then:

(x-5) is negative

(x+2) is positive

Then (x+2)(x-5) is negative, so it's undefined.

So the function is defined for x>5 and x<-2

So the correct option is option D.