For f(x)=1/x-2 and g(x)=√x, find g(f(x)) and state its domain

Answer 1

First of all, please bear with me as I attempt to type this explanation in a way that you hopefully can follow and understand! Also, understand that in composition of functions, one of these functions is the input. Since you are asked to find g(f(x)), the input function is f(x), which means you are going to replace every 'x' in g(x) with 'f(x)'. Since g(x) = √x, then g(f(x)) = √f(x), which gives you g(f(x)) = √1/x-2 (all under the square root). That simplifies to g(f(x)) = 1/√x-2 and since square roots are generally not allowed in denominators, you have to rationalize the denominator (multiply the 1/√x-2 by √x-2/√x-2). This will give your answer of g(f(x)) = (√x-2)/(x-2).

As far as the domain goes, there are two parts to look at that will affect the domain. First of all, the number inside of the square root cannot be negative, or else you get into imaginary numbers. With that said, every value of x that is less than 2 will give you a negative number inside the square root. Therefore, x has to be greater than 2. But, look at the denominator and remember that a zero in the denominator will make a function undefined. Therefore, x cannot equal 2. So your domain is every number greater than 2, to not include 2. In interval notation, this will be Domain: (2, ∞).

Like I said, I hope this is clear enough for you to understand. If there is any part of this explanation that's not clear to you, let me know and I'll try to clarify :)