How do we find the range of the binary relation #4x^2 + 9y^2 = 36# ?

2 Answers
Aug 13, 2017

Set x equal to zero and then solve the equation for both values of y; the negative value will be the minimum of the range and the positive value will be the maximum.

Explanation:

Given: #4x^2 + 9y^2 = 36#

Let #x = 0#:

#4(0)^2 + 9y^2 = 36#

#9y^2=36#

#y^2=4#

#y = -2 and y = 2#

The range is #-2 <= y <= 2#

Aug 13, 2017

The range is #y in [-2,2]#

Explanation:

The equation is

#4x^2+9y^2=36#

#4x^2=36-9y^2#

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

#x=3/2sqrt(4-y^2)#

#x=3/2sqrt((2-y)(2+y))#

#x=f(y)#

#x# will depend on #(2-y)(2+y)>=0#

Let #p(y)=(2-y)(2+y)#

We need a sign chart

#color(white)(aaaa)##y##color(white)(aaaa)##-oo##color(white)(aaaaa)##-2##color(white)(aaaaaaaa)##2##color(white)(aaaaa)##+oo#

#color(white)(aaaa)##2+y##color(white)(aaaaaa)##-##color(white)(aa)##0##color(white)(aaaa)##+##color(white)(aaaaa)##+#

#color(white)(aaaa)##2-y##color(white)(aaaaaa)##+##color(white)(aaaaaaa)##+##color(white)(aa)##0##color(white)(aa)##-#

#color(white)(aaaa)##p(y)##color(white)(aaaaaaa)##-##color(white)(aa)##0##color(white)(aaaa)##+##color(white)(aa)##0##color(white)(aa)##-#

Therefore,

#p(y)>=0# when #y in [-2,2]#

graph{4x^2+9y^2=36 [-9.22, 8.56, -3.235, 5.654]}