Find the Function (fog)(x)

Date: 07/02/2003 at 19:44:55
From: Pamela
Subject: Functions

Let f(x) = 2x-1 and g(x) = (x+5)/2. Find the function (fog)(x). 
Simplify your answer.

   g(x) = (x+5)/2
        = 2[(x+5)/2]-1
        = x+5-1
        = x+4

Date: 07/03/2003 at 11:02:27
From: Doctor Mike
Subject: Re: Functions

Pamela,
  
Your final answer is right. But there is one thing I should mention 
about the way you wrote down the steps leading to that final answer.  
  
Notice that you have the correct equation for g(x), and then right 
after that is an equals sign. If you think about that for a moment you  
will probably see what is wrong with that. You clearly understand
what you are doing here, since you carry through to the correct final 
answer, BUT when you put the "=" right at the start of that second 
line, it looks as if you are saying that the first line EQUALS the 
second line. That is not what you meant, I'm sure.
  
So how to fix it?  Just make sure that the equations and expressions 
you write down really do say the things you are thinking. Here is the 
way I would do it. 

  g(x)=(x+5)/2
  fog(x) = f(g(x))
         = f((x+5)/2)
         = 2[(x+5)/2]-1
         = x+5-1
         = x+4

Do you see the important difference? When I start out the second line 
with "fog(x)=" I am making it clear that I am starting out with the 
next step. The first step was to write down what "g" means. Then with 
that done I start on the main part, which is to get an equation for 
fog. It is very clear which part of what I'm writing has to do with g 
and which part has to do with fog.
  
You might be thinking, "Well, Dr. Mike, you also wrote down a bunch of 
things connected by the = sign." Yes, I did, but with what I wrote 
down, all the things connected by = really ARE all equal to each 
other. The only objection I have to what you wrote down is that single 
= you used to connect the expression for g and the expression for
fog. This is a common thing to happen when you are eager to get your 
thoughts written down. Just make sure what your fingers write down is 
the same as what your mind is thinking.   

Thanks for writing to Dr. Math.

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