so if f(g(x)) exists, then the domain of f(g(x)) is always the domain of g(x)
Am I correct?
Yes, this is the convention adopted in Methods. For f • g to be defined, we require that the range of g is a subset of the domain of f. When this is true, the domain of f • g will also be the domain of g.
Metaphorically, when composing f with g, the permissible "inputs" to f come from the "outputs" from g. Hence, if every "input" to g results in an "output" that can be "input" to f, it follows that every "input" to g is also an "input" to f • g.
Looking at it from the other side, there may be values within the range of g that can not be "input" to f. Hence, any "input" to g that results in an "output" which can't be "input" to f can't be an input to f • g.
IDK tbh, but test out the theory by using these two functions.
=\sqrt x \\<br />\\<br />g(x) = x^2)
I just tried to answer your first question by using these as an example... and then I absolutely baffled myself.
Common sense says that the graph of the composite should just be the graph of y=x, but the domain is still very much affected.
Check it out on Desmos.
Common sense says no such thing.
 = \sqrt{x^2})
is always positive (in fact, this is just the absolute value function |x|), hence it can not be the case that f(x) = x.
However, the maximal
domain of
is all real numbers, because the range of
 = x^2)
is a subset of the maximal domain of
 = \sqrt{x})
, and the maximal domain of g(x) is all real numbers.
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