What is the domain and range of #y =sqrt(x-3) - sqrt(x+3)#?

3 Answers
Jul 2, 2018

Answer:

Domain: #[3, oo) " or " x >= 3#

Range: #[-sqrt(6), 0) " or " -sqrt(6) <= y < 0#

Explanation:

Given: #y = sqrt(x-3) - sqrt(x + 3)#

Both the domain is the valid inputs #x#. The range is the valid outputs #y#.

Since we have two square roots, the domain and the range will be limited.

#color(blue)"Find the Domain:"#

The terms under each radical must be #>= 0#:

#x - 3 >= 0; " " x + 3 >= 0#

#x >= 3; " "x >= -3#

Since the first expression must be #>=3#, this is what limits the domain.

Domain: #[3, oo) " or " x >= 3#

#color(red)"Find the Range:"#

The range is based on the limited domain.

Let #x = 3 => y = sqrt(3-3) - sqrt(3+3) = -sqrt(6)#

Let #x = 100 => y = sqrt(97) - sqrt(103) ~~-.3#

Let #x = 1000 => y = sqrt(997) - sqrt(1003) ~~-.09#

#x -> oo, y -> 0#

Range: #[-sqrt(6), 0) " or " -sqrt(6) <= y < 0#

Jul 2, 2018

Answer:

The domain is #x in [3,+oo)#. The range is #y in [-sqrt(6),0^-)#

Explanation:

What's under the #sqrt# sign must be #>=0#

#=>#, #x-3>=0# and #x+3>=0#

#=>#, #{(x>=3),(x>=-3):}#

Therefore,

The domain is #(x>=3)nn(x>=-3)#

That is, #x in [3,+oo)#

When #x=3#, #=>#, #y=0-sqrt6#

And when #x->+oo#

#lim_(x->+oo)y=0^-#

Therefore,

The range is #y in [-sqrt(6),0^-)#

graph{sqrt(x-3)-sqrt(x+3) [-1.42, 18.58, -6.36, 3.64]}

Jul 2, 2018

Answer:

Domain: #[3, oo)#

Range: #[-sqrt(6), 0)#

Explanation:

Given:

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

First note that the square roots are well defined and real if and only if #x-3 >= 0# and #x+3 >= 0#. Hence it is necessary and sufficient that #x >= 3#.

So the domain of the function is #[3, oo)#

To find the range, note that when #x = 3# then:

#y = sqrt((color(blue)(3))-3)-sqrt((color(blue)(3))+3) = sqrt(0)-sqrt(6) = -sqrt(6)#

We find:

#lim_(x->oo) (sqrt(x-3)-sqrt(x+3)) = lim_(x->oo) ((sqrt(x-3)-sqrt(x+3))(sqrt(x-3)+sqrt(x+3)))/(sqrt(x-3)+sqrt(x+3))#

#color(white)(lim_(x->oo) (sqrt(x-3)-sqrt(x+3))) = lim_(x->oo) ((x-3)-(x+3))/(sqrt(x-3)+sqrt(x+3))#

#color(white)(lim_(x->oo) (sqrt(x-3)-sqrt(x+3))) = lim_(x->oo) (-6)/(sqrt(x-3)+sqrt(x+3))#

#color(white)(lim_(x->oo) (sqrt(x-3)-sqrt(x+3))) = 0#

Note that #-6/(sqrt(x-3)+sqrt(x+3))# is continuous and monotonically increasing.

Hence the range of the given function runs from the minimum value #-sqrt(6)# up to but not including the limit #0#.

That is, the range is #[-sqrt(6), 0)#

graph{y = sqrt(x-3)-sqrt(x+3) [-10, 10, -5, 5]}