Htykuut

I have two rather simple functions. I don't know how to find that the dominator is different than 0. Find the domain of functions:

Function 1:

$f(x) = frac{4-x}{arcsinfrac{x}{4}}$

Assumption 1:

$-1leq arcsinfrac{x}{4} leq 1$

$-1leq frac{x}{4} leq 1$

$-4leq x leq 4$

$xin<4;4>$

Assumption 2:

$arcsinfrac{x}{4} neq 0$

Here, I have no idea how to proceed further with assumption 2.

Function 2:

$f(x) = frac{sqrt{2x-1}}{2+arccosfrac{x+1}{4}}$

Assumption 1:

$sqrt{2x-1} geq 0$

$2x geq 1$

$x geq frac{1}{2}$

Assumption 2:

$ -1 leq arccos frac{x+1}{4} leq 1 $

$ -1 leq frac{x+1}{4} leq 1 $

$ -4 leq x + 1 leq 4 $

$ -5 leq x leq 3 $

Assumption 3:

$ 2 + arccosfrac{x+1}{4} neq 0 $

Here once again, no idea how to proceed further.

Would anyone be able to help me understand these examples, how to find that the denominator is not equal to $0$, when there's arccos or arcsin in the denominator? Thanks! I do hope my tags are okay...

For the first function, the fraction is defined when $arcsin frac{x}{4} ne 0 implies frac{x}{4} ne sin 0=0 implies xne 0$

So the domain of the first function is $[-4,4]-{0}$

For the second function $$2+arccos frac{x+1}{4} ne 0$$
$$implies arccos frac{x+1}{4} ne -2$$
Apply cos both sides,
$$implies frac{x+1}{4} ne cos(-2)$$
$$implies x ne 4cos(-2) -1$$