Find the domain and range of this equation: It is a square root equation.?

The domain is easy.  The square root is the only operation that restricts the domain, requiring that  -(x + 20) under the square root must be 0 or more:

    -(x + 20) >= 0

     x + 20 <= 0    . . . . multiply by -1, reversing inequality

     x <= -20         . . . . and subtract 20

So the domain is (-oo, -20] in interval notation.

For the range, the easy way is to see that any non negative number can be a result of just the square root part.  So:

    √[-(x + 20)] >= 0                  . . . . with all values possible

    13√[-(x + 20)] >= 0              . . . . multiply both sides by 13

    13√[-(x + 20)] - 41 >= -41    . . . . subtract 41 from both sides

The left side is g(x) now, and you get g(x) >= -41 as your range, or [-41, +oo) in interval notation.

As far as "isolating y", there's no y in the problem.  If you're finding the inverse of g, it's common to let y = f(x) and then solve.  That doesn't do much here, but the process is basically reversing the operations used to turn x into y.

    y  =  g(x)

    y  =  13√[-(x + 20)] - 41

    y + 41  =  13√[-(x + 20)]

    (y + 41) / 13  =  √[-(x + 20)]

At this point, note that the left side must be 0 or greater in order to be the result of a square root.  That gives you the y >= -41 result above.  To finish and get the inverse:

    [(y + 41) / 13]²  =  -(x + 20)

    -[(y + 41) / 13]²  =  x + 20

    -20 - [(y + 41) / 13]²  =  x

And that's another way to see that the domain of x is x <= -20, since the quantity in [] brackets can be any non-negative number; and so can its square.

f(x) = √(x)   

has domain x>= 0  

and  range  f(x) => 0 

You're problem is confusing 

Do you want the domain and range of 

g(x) and the inverse of g(x)  

so the 

domain of g(x)  

requires that -(x+20) >=0  

-x -20 >=0   

subtract x from both sides 

-20>=  x

g(x) 

Domain:  

x<= -20 

so range of  

√( -(x+ 20) )     >=   0   

13* x>=    is still >=    0 

subtract  

range -41  >=

y>= -41  

For the inverse of g(x) 

to find the domain and range, you don't have

to actually domain what the equation is 

Since the domain and range are just 

the reverse of the original equation. 

just reverse the domain and range 

g^(-1)(x)  

domain  x>= -41   

range   y or  g^(-1)(x)  <= -20  

If you want to find the equation 

square both sides 

swap  x and y 

x = 13*sqrt( -(y+20)) - 41  

add 41 to both sides 

x+41 = 13*sqrt( -(y+20)) 

(x+41)^2  = -169(y+20)  

(x+41)^2  = -169y - 3380

169y =  -(3380 + (x+41)^2) 

y = -(3380 + (x+41)^2) / 169   

the domain is limited to 

domain x>=-41