One way of combining two different functions is
to form the composition of one with the other. The domain of this is the set of all
x in the domain of
g such that
g(
x) is in the domain of
f. The composition of the function
f with
g is:
For example:
The domain is then
When determining the domain of a composite function, you need to restrict the outputs of
g so that they are in the domain for
f. For example, to find the domain of
f composed with
g of
x given that
f(
x)=1/
x and
g(
x)=
x+1, note the outputs of
g. They can be any real number, but the domain of
f is restricted to all real numbers except 0. Therefore, the outputs of
g must also be restricted to all real numbers except 0. This means that in
g(
x)=
x+1
≠ 0, or
x ≠ -1. So, the domain of
f composed with
g of
x is all real numbers except
x=-1.
Next is to find the domain of the domain of
f composed with
g of
x for the functions:
It may look like the domain of the composition is the set of all real numbers. Although, this is incorrect because the domain of g is:
Thus, the domain of f composed with g of x is also:
Finally, we can identify a composite function by expressing a function as a composition of two functions. We may not know f(x) or g(x) at first:
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