Composition Functions with Added x Value

Date: 05/13/2001 at 13:21:44
From: Eileen Gonzalez
Subject: Composition functions with added "x " value

Hello,

I would really appreciate some direction on this question. It throws 
me off because it adds another value for x. How to plug this in?

     f(x) = 5x  and 
     g(x) = 2x^2-x+1

The question reads:

     If x = 1, evaluate g(f(f(x)))

I'm confused with this added value of x = 1.

What I did to solve was:

   f(5(5*1))
   f(25*1)
   2(25*1)^2-25*1+1 =
   2*625-25+1 =
   1250-25+1 = 1226

Is this right? I feel as if I'm getting off track with x = 1, but 
can't be sure.

Many thanks,
Eileen

Date: 05/13/2001 at 23:11:09
From: Doctor Peterson
Subject: Re: Composition functions with added "x " value

Hi, Eileen.

I don't know why you started with f(5(5*1)); that would be f(f(f(1))). 
Ah - you must have meant g(5(5*1)), which is what you want to do. And 
you did it right.

There are two ways to do this kind of problem. You can either start by 
determining the composition g(f(f(x))) in general, and then plugging 
in the given value of x; or you can find g(f(f(1))) specifically. You 
did the latter. Let's do it the longer way and think about what this 
all means.

     f(x) = 5x
     g(x) = 2x^2 - x + 1

     f(f(x)) = f(5x)
             = 5(5x)
             = 25x

     g(f(f(x))) = g(25x)
                = 2(25x)^2 - (25x) + 1
                = 1250x^2 - 25x + 1

Now we can plug in x = 1, and we get

     g(f(f(1))) = 1250(1)^2 - 25(1) + 1 = 1226

This separates out the two things that are happening: composition of 
functions, and evaluation for a specific value.

What are we doing? The composition of functions can be seen as hooking 
up three machines together:

     x --> f --> f --> g --> ?

Your method (which is the quicker way) just feeds 1 into the pipeline, 
and collects the 1226 that drips out the other end.

My method (which would be better if you had to do the work for 
different values of x) analyzes the machine and replaces it with a 
single machine that does the same thing:

     x --> ffg --> ?

Then when I'd finished all the plumbing, I put a 1 into the input and 
got my 1226 out the end. A lot of work for one drop of output, but it 
helped me understand the process better, and that's what composition 
of functions is really about: this concept that you have not just a 
single value to think about, but that the whole pipeline is really a 
whole new function that can be thought of just the same way you think 
of any other function. Rather than think of a series of things that 
happen to one value (your approach), we think of a single bigger thing 
that can happen to any value (mine). The fact that they give the same 
answer is what we're after.

I hope that helps.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/14/2001 at 14:49:08
From: Eileen JQ Gonzalez
Subject: Re: Composition functions with added "x " value

Dr. Peterson,

You just made my day! Thank you so much for your prompt response! 
You've definitely cleared a few things up for me.

Eileen.