How to find domain for #f(x) = sqrt(25 - x^2)#?

1 Answer
Sep 29, 2015

Answer:

#[-5,5]#

Explanation:

So that the answer won't be imaginary, #25-x^2# must be greater than or equal to #0#.

#{x|25-x^2>=0}#
#{x|25-x^2-25>=0-25}#
#{x|-x^2>=-25}#
#{x|(-1)(-x^2)<=(-1)(-25)}#
#{x|x^2<=25}#

This will then be separated to two cases:
If #x# is positive:
#{x|x^2<=25}#
#{x|x<=5}#
#color(blue)((-oo,5])#

If #x# is negative:
#{x|x^2<=25}#
#{x|x>=-5}#
#color(red)([-5,oo))#

We will then get the intersection of the two intervals:
#color(blue)((-oo,5])nncolor(red)([-5,oo))#
#color(magenta)([-5,5]#