algebra

how to find a domain of a combined function? example f(g(x)) do we have to consider the domain of both f and g?

I think so

I am going to experiment with a specific example.
g(x) = 1/x
from g(x) x cannot be 0

f(x) = 3/(x+2)
from f(x) x cannot be -2

f(g(x))= 3/ (1/x +2)
x can't be 0
1/x can't be -2 therefore x can't be -1/2
I don't think that x can be -2 either.


But what if i simplify first,
f(g(x))= 3/ (1/x +2)
f(g(x))= 3/ {(1+2x)/x}
f(g(x))= 3 * x/(1+2x)
now x can't be -1/2 (x=-2 or 0 would work here)

anyway, I think x cannot be -2, -1/2 or 0
I will get back to you on this one.

Okay, I have consulted an expert and this is my feedback.

I am going to experiment with a specific example.
g(x) = 1/x
from g(x) x cannot be 0

f(x) = 3/(x+2)
from f(x) x cannot be -2 (THIS IS NOT RELEVANT)

f(g(x))= 3/ (1/x +2)
x can't be 0
1/x can't be -2 therefore x can't be -1/2

So, x is not defined at -1/2 or 0 (but it is defined at -2)

So the restrictions are from g(x) and from f(g(x)) but not from f(x)

this was another more succinct answer.

http://gyazo.com/0f99cac55518414ad8b93eb7d5c9b5c4