How do you express the function #h(x)=(x + 3)^6# in the form f o g?

Question

16698 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-12-29T08:45:57

This can be done in an infinite amount of ways. However, the easiest way is to use the sixth power to your advantage.

#f@g# means that #g(x)# is being plugged INTO #f(x)#, which means that #f(x)# will have the sixth power in it.

If #h(x)=(x+3)^6#, #h(x)=(f@g)(x)# if

#f(x)=x^6#
#g(x)=x+3#

Work backwards: to find #(f@g)(x)#, take #g(x)# and put it inside of #f(x)#--you get #(f@g)(x)=(x+3)^6=h(x)#.

Other variations include:

#f(x)=sqrtx#
#g(x)=(x+3)^12#
#(f@g)(x)=(x+3)^6#

#f(x)=x+729#
#g(x)=x^6+18 x^5+135 x^4+540 x^3+1215 x^2+1458 x#
#(f@g)(x)=(x+3)^6#