Expert: Paul Klarreich - 1/17/2008

Question
Hi!
Im just having trouble with one of my math problems...it says:
Suppose that
     g(x)=2x+1
     h(x)=4x^2+4x+7
Find a function 'f' such that f (little o)g =h. (Think about what operations you would have to perfor on the formula for 'g' to end up with the formula for 'h').
Now suppose that:
      f(x)=3x+5
      h(x)=3x^2+3x+2
Use the ssame sort of reasoning to find a function 'g' such that f (little o) g=h.

Thanks! I tried to type the problem as clearly as i could.

Answer
Questioner:   Carly
Category:  Advanced Math
Private:  No
 
Subject:  Algebra..functions
Question:  Hi!
Im just having trouble with one of my math problems...it says:
Suppose that
    g(x)=2x+1
    h(x)=4x^2+4x+7
Find a function 'f' such that f (little o)g =h. (Think about what operations you would have to perfor on the formula for 'g' to end up with the formula for 'h').
Now suppose that:
     f(x)=3x+5
     h(x)=3x^2+3x+2
Use the ssame sort of reasoning to find a function 'g' such that f (little o) g=h.

Thanks! I tried to type the problem as clearly as i could.
..............................................
Hi, Carly,

You can write  f(g(x)) = h(x).  Your 'little o' is just the 'composition' of f and g.

Now if  h(x) = 4x^2 + 4x + 7 and you want it to be  f(2x + 1), for some f, why not do long division first?

2x + 1 into 4x^2 + 4x + 7 gives  2x + 1
           4x^2 + 2x
           -----------
                  2x + 7
                  2x + 1
                 --------
                       6, the remainder.

So 4x^2 + 4x + 7 = (2x + 1)^2 + 6

=  g(x)^2 + 6.

So your f(x) = x^2 + 6.

Try it out and you'll see.


f(x)=3x+5
     h(x)=3x^2+3x+2

3x+5 into 3x^2+3x+2 is x - 2/3
         3x^2 +5x
         -----------
              - 2x + 2
              - 2x -10/3
            ----------------
                   16/3

h = (3x + 5)(x - 2/3) + 16/3
h = (g)(x - 2/3) + 16/3

Now what about x - 2/3? That is (3x - 2)/3 = (3x + 5 - 7)/3 = (g - 7)/3

h = (g)((g - 7)/3) + 16/3

That is your function.  

f(x) = x(x - 7)/3 + 16/3

or:
   x^2 - 7x + 16
 = --------------
        3
.........................................
YOU say you're not convinced.  Here is the check:

      (3x+5)^2 - 7(3x+5) + 16
f(g) = --------------------------
            3

      9x^2 + 30x + 25 - 21x - 35 + 16
f(g) = -------------------------------
            3

      9x^2 + 9x + 6
f(g) = --------------- = 3x^2 + 3x + 2
            3