Help with finding ranges of composite functions

[BigZero1]

I seem to have a lot of trouble with ranges of combined functions. I must be missing something or not understanding something. Domains don't give me a problem but when it comes to ranges I usually end up having to resort to a graph and plotting points etc and I'm wondering if there is a better way. I'm wondering if someone can guide me through the thought process of finding the range for the one given below.



Domain: I found the range as: Even if that's the range, I think the process I use to find the range is not correct so I'm wondering about other people's way of finding the range. Also, is there a common way to find ranges of most combined functions or is it usually different.

 

[Steven G]

Let k be any real number. Is k in the rage? Only if there is an x value such that f(x) = k.

Sqrt((x+1)/x) = k. Right away we see that k>=0

(x+1)/x = k^2

x+1 = k^2x
1 = (k^2-1)x
x= 1/(k^2-1) which is not defined if k = +/-1. The domain is [0,1) U (1,oo)

To see this in action: Is 5 in the range? Just let x =1/(5^2-1) = 1/24. Now sqrt[ (1/24 +1)/(1/24)] = sqrt[25] = 5. So yes, 5 is in the range.

 

[Steven G]

Can we see your method so we can inform you if your process works?

 

[BigZero1]

My work is too much of a disorganized incomprehensible mess to bother looking at. I really had no method. It was a combination of plugging in values and trying to graph. I figured there would be a better way.
I applied the method you showed me to a few problems involving ranges and am very happy with the results. Thanks so much!!!