How can i find the Domain?

#f(x)= sqrt(1-x^2) + tan(πx)#

1 Answer
Firelight
Feb 1, 2018

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)

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