Question

How can i find the Domain?

`f(x)= sqrt(1-x^2) + tan(πx)`

Answer 1

X≠0.5, -0.5
-1<X<1

Explanation:

separate the terms first

`sqrt(1-x^2)`

The value under a square root sign cannot be negative or it will become an imaginary number which means:
`1-x^2` cannot be less 0
so if we find the value where:
`1-x^2=0`
we know what `x` cannot be less than
so move the `-x^2` to the other side to have:
`x^2=1`
square root both sides:
`x=sqrt(1)=1 and -1`
so x cannot be less than negative one, or more than positive one
then for the next term
`tan(pix)` I'm going to pull the `pi` outside the brackets so it's easier to realize that it is `b` in the general formula that applies to tangent:
`atanb(x+c)+d`
this is important because to find the period (in this case the distance between asymptotes) you need to do:
`pi/b`
so in
`tanpi(x)`
`b=pi`
`period=pi/pi=1`
so the asymptotes are
`x= 0.5 and -0.5`
(I got this because this tangent graph has no been moved so it's center will be at `(0,0)` and if the distance between asymptotes is 1 then move half of 1 in either direction to have 0.5 or -0.5)