How do you write # y = 3sqrt(1 + x^2)# as a composition of two simpler functions?

2 Answers
Aug 11, 2015

Define these functions:
#g(x)=1+x^2#
#f(x)=3sqrtx#

Then:
#y(x)=f(g(x))#

Aug 13, 2015

There is more than one way to do this.

Explanation:

Adrian D has given one answer, here are two more:

Let #g(x)# be the first thing we do if we knew #x# and started to calculate:

#g(x) = x^2" "#

Now #f# will be the rest of the calculation we would do (after we found #x^2#)

It may be easier to think about if we gave #g(x)# a temporary name, say #g(x)=u#

So we see that #y = 3sqrt(1+u)#

So #f(u) = 3sqrt(1+u)# and that tells us we want:

#f(x) = 3sqrt(1+x)#

Another answer is to let #f(x)# be the last thing we would do in calculating #y#.

So let #f(x) = 3x#

To get #y = f(g(x))# we need #3g(x) = y#

So let #g(x) = sqrt(1+x^2)#