# Decomposition of Functions

1. May 28, 2012

### Bashyboy

1. The problem statement, all variables and given/known data
If $g(x) = 2x + 1$ and $h(x) = 4x^2 +4x + 7$, find a function f such that
$f o g = h$

2. Relevant equations

3. The attempt at a solution

Well, I know I have to decompose h into two parts: the give one being g, and the other f. But I can't seem to perceive any relation between g and h, how can I find the pattern?

Last edited: May 28, 2012
2. May 28, 2012

### Staff: Mentor

To start, complete the square in 4x^2 + 4x + 7. You might find the relationship you need after doing this.

3. May 28, 2012

### Infinitum

Hi Bashyboy!

Here's how you can start.

By common sense(and looking at the question carefully ), you'll see f(x) should be a quadratic equation. Assume it to be any general quadratic equation with variable coefficients.

Now you need to the function f(x) such that,

$f(g(x)) = h(x)$

4. May 28, 2012

### Bashyboy

I completed the square, but I still don't seem to see a connection.

5. May 28, 2012

### HallsofIvy

Staff Emeritus
As Infinitum says, f must clearly be a quadratic, say $f(x)= ax^2+ bx+ c$ so that $f(2x+1)= a(2x+1)^2+ b(2x+1)+ c= 4x^2+ 4x+ 7$. Multiply the left side out and you have three equations for a, b, and c.

Mark44's suggestion, completing the square, works with a little "massaging".
$4x^2+ 4x+ 7= 4(x^2+ x+ (1/4)- (1/4))+ 7= 4(x^2+ x+ 1/4)+ 6= 4(x+ 1/2)^2+ 6$

Now, 2(x+ 1/2)= 2x+ 1 so we have to, somehow, get a "2" into that square. We do that, of course, by multiplying that 4 back into the square:
$4x^2+ 4x+ 7= (2x+1)^2+ 6$.

Last edited: May 28, 2012
6. May 28, 2012