Sorry Ron but I don't follow your steps.
You said:
(1) g(x)=1+x^2, f(x)=x^(3/2), then f(g(x))=(1+x^2)^(3/2)...Where did f(x) and g(x) come from? How did you get x^(3/2)? Where did f(g(x)) come from?
(2) g(x)=2x^2+3, f(x)=|x|, then f(g(x))=|2x^2+3|...Where did the |x| come from? How does f(g(x)) = |2x^2 + 3|.
How do question 1 and 2 equate to H?
The come from an examination of the target function. You want
f(g(x))=(1+x^2)^(3/2),
so you need to look for an inner and outer function to decompose the
composite function into. Here we have something raise to a power, so
raising to a power will do for f, and what is raised to the power will do
for g. There is no method other than examination of the structure of the
required form of the composite function.
Also the decomposition is not unique. For example I could have chosen
g(x)=(1+x^2)^3, and f(x)=x^(1/2). In fact subject to some pretty lose
restrictions g(x) could be almost anything, but once chosen determines
f(x)
As before, choose the simplest functions that will do. Look at the target(2) g(x)=2x^2+3, f(x)=|x|, then f(g(x))=|2x^2+3|...Where did the |x| come from? How does f(g(x)) = |2x^2 + 3|.
and it should suggest suitable functions.
H is defined in the question to be the composite function, so:How do question 1 and 2 equate to H?
H(x)=f(g(x)).
It is just another name for f(g(x)) or f o g.
RonL