Good evening Candice! I will look at this later when I am ready

Sure sure. Thanks Mr Eric :)

Candice, it is noteworthy to know that the range of gf(x) will be equal to the range of g(x) given its input (which is going to come from the output of f(x)).

We say that Rgf = Rg.

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As long as the output of f(x) is a subset of (i.e. is fully contained in) the list of possible inputs of g(x), the composite function will exist.

We say that Rf ⊆ Dg.

Note that every value of the range of f must also be present in the domain of g (hence, the "subset" stipulation).

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In this particular case, our approach is to first find the range (output) of f(x), which upon closer inspection will be all possible outputs except for y = -1.

Since g(x) is generous enough to allow any value of x into it (shown as x ϵ R), it's pretty obvious that the range of f(x) is going to be a (proper) subset of the domain (input) of g(x). So, the composite function gf(x) exists.

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The range of f(x) (all values except for -1) will enter the function g(x) as a domain.

Whatever comes out of g(x) as the output from this input will become its domain, which we can easily work out to be all values except for 3.