Decomposing Functions: Finding the Pattern

[Bashyboy]

Homework Statement

If $g(x) = 2x + 1$ and $h(x) = 4x^2 +4x + 7$, find a function f such that
$f o g = h$

Homework Equations

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?

 

[Mark44]

Homework Statement

If $g(x) = 2x + 1$ and $h(x) = 4x^2 +4x + 7$, find a function f such that
$f o g = h$

Homework Equations

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?

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

 

[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,

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

 

[Bashyboy]

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

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

 

[HallsofIvy]

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:
$4x^2+ 4x+ 7= (2x+1)^2+ 6$.

 

[sponsoredwalk]

But I can't seem to perceive any relation between g and h, how can I find the pattern?

Ask yourself how you can turn g(x) into h(x), what do you need to do to g(x) to turn it into h(x)? Whenever you do something to g(x) you an interpret it as composing g with some function f, for example if you want to add 1 to g(x) you can mathematically express this idea as (fog)(x) = (2x + 1) + 1 & then use this information to find f. So what does this say about the function f? It has to be the function f(x) = x + 1 so that you have (fog)(x) = f(g(x)) = [g(x)] + 1 = (2x + 1) + 1. Now do whatever you need to do to turn g(x) into the h(x) given in your problem & then use what you've done to find f.