What is the DOMAIN of f(g(x)) if...?

Question

f(x)= x+1
-and-
g(x)=radical of x-3

please help me!
i hate domain and range!

Answer 1

ok. since you are taking f(g(x)), the domain of f will depend on the domain of g(x) and f(x). Domain is the x values for which f(x) exists. So, in this case, the domain of g(x) is:
[3, infinity), since we can not take the square root of numbers less than 0.
The domain of f(x) is:
(-infinity,infinity) since f is defined for all x.
Now, we want to see where these domains overlap. By looking at the domains of [3, infinity) and (-infinity,infinity), we see that [3, infinity] is where they overlap. So, the domain of f(g(x)) = domain of g(x) = [3, infinity)

Now, to get f(g(x)), just plug in g(x) for x in the function f(x).
So, x= radical of (x-3). Plug that into f(x) = x + 1 to get:

f(g(x)) = radical (x - 3) + 1

Now, we know that this function is continuously increasing, since the square root function is continuously increasing.
Thus, the minimum value of f(g(x)) will occur at x=3, since that is the first point where f(g(x)) exists.

f(3) = radical ((3) - 3) + 1 = 0 + 1 = 1

Now, since f(g(x)) is continuously increasing, it's range will go on to infinity.

Thus, the range of f(g(x)) = [minimum value of f(g(x)), maximum value of f(g(x))] = [1, infinity )

Answer 2

f(g(x)) would equal radical(x-3) + 1
Since you can't take the square root of a negative number, x must be greater than or equal to 3 (that would be the domain). Plug that back into the equation and you get that y has to be greater than or equal to 1 (that's the range). Hope that helped :)