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

1 Answer
Nov 24, 2015

#f(g(x))=xcolor(white)("XXX")#and#color(white)("XXX")g(f(x))=x#
Both functions and their compositions have Domains and Ranges of #(-oo,+oo)#

Explanation:

Sometimes the use of #x# in multiple definitions can cause confusion, so let's re-write the base equations as:
#color(white)("XXX")f(a)=8a#
and
#color(white)("XXX")g(b)=b/8#

So, replacing #a# with #g(b)# we have
#color(white)("XXX")f(g(b)) = 8*g(b) = 8*b/8 = b#
(or, using the original #x# as the variable: #f(g(x))=x#)

Similarly #g(f(x)) = x#

#{ (f(x) = 8x), (g(x) = (x)/(8)), (f(g(x))=x), (g(f(x)))=x :}#
#color(white)("XXX")#are all defined for all values of# x#
and
therefore have Domains of all Real values, #(-oo,+oo)#

Replacing, for example #f(x)# with #y#
we can see that #f(x)=8x <=>y/8=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 #(-oo,+oo)#