How do you find the domain and range of #2/ root4(9-x^2)#?

1 Answer
Dec 16, 2017

Answer:

The domain is #x in (-3,3)#. The range is #y in [1.155, +oo)#

Explanation:

Let #y=2/root(4)(9-x^2)#

The denominator must be #!=0# and #>0#

Therefore,

#9-x^2>0#

#x^2<9#

#x in (-3,3)#

The domain is #x in (-3,3)#

To find the range, proceed as follows

#y^4=16/(9-x^2)#

#9-x^2=16/y^4#

#x^2=9-16/y^4#

#x=sqrt((9y^4-16)/(y^4))#

Therefore,

#y!=0#

and

#9y^4-16>=0#

#y^4>=16/9#

#y>=2/sqrt3#

The range is #y in [2/sqrt3, +oo)#

graph{root(4)(16/(9-x^2)) [-7.02, 7.03, -2.197, 4.827]}