Let f(x)=8x and g(x)=x8, how do you find each of the compositions and domain and range?

1 Answer
Nov 24, 2015

f(g(x))=xXXXandXXXg(f(x))=x
Both functions and their compositions have Domains and Ranges of (,+)

Explanation:

Sometimes the use of x in multiple definitions can cause confusion, so let's re-write the base equations as:
XXXf(a)=8a
and
XXXg(b)=b8

So, replacing a with g(b) we have
XXXf(g(b))=8g(b)=8b8=b
(or, using the original x as the variable: f(g(x))=x)

Similarly g(f(x))=x

{(f(x)=8x),(g(x)=x8),(f(g(x))=x),(g(f(x)))=x
XXXare all defined for all values ofx
and
therefore have Domains of all Real values, (,+)

Replacing, for example f(x) with y
we can see that f(x)=8xy8=x
which is defined for all Real values of y.
(this process can be done for all of the functions)

So the Ranges of these functions is also all Real values (,+)