# Thread: component of composite function: given fcns (f o g)(x) and g(x), how to find f(x) ?

1. Originally Posted by hellawowser
(f o g)(x) is 2x^2 -4x + 1 and g(x) = 2x - 3. That means f(2x - 3) = 2x^2 -4x + 1 right? How do i get to know the f equation?
Here the idea is to express the given function in terms of (2x-3). So

2x^2 -4x + 1 = 1/2 * (4x^2 -8x + 2) = 1/2 * [(4x^2 -12x + 9) + 4x - 7] = 1/2 * [(2x-3)^2 + 2(2x -3) - 1]

continue.....

2. Originally Posted by hellawowser
f(x) = 1/2x^2 + x - 1/2 ??
You can check!

Using your candidate for f(x), determine the composite f(g(x)).

Does the simplified result match the given?

3. Originally Posted by hellawowser
I've meant that when you only know (f o g)(x) you have several pairs of equation for f(x) and g(x) , but when you know one of them, you still can have a lot of possibilities for the other one?

Like i said i've only tried to fill in the equation : f(2x - 3) = 2x^2 -4x + 1 but i couldnt operate with it.
Recently i've tried to analysis the domain and range for each but also couldnt get anything useful from it.
I observe that no one has suggested what to me is the best algebraic approach to this (no guessing).

You know that f(2x - 3) = 2x^2 - 4x + 1, and you want to find an expression for f(u). To do that, you can just substitute
u = 2x - 3

and solve to express x in terms of u. Put this in place of x on the right side, and simplify to obtain f(u).

Another way to describe this process, if you are comfortable with the notation of function composition, is that you want $f \circ g = h$, where h(x) = 2x^2 - 4x + 1 and g(x) = 2x - 3, but you don't know f. You can just compose each side with $g^{-1}$:

$(f \circ g) \circ g^{-1} = h \circ g^{-1}$

The left side simplifies to f, so the right side is the answer.

4. Originally Posted by Dr.Peterson
I observe that no one has suggested what to me is the best algebraic approach to this (no guessing).

You know that f(2x - 3) = 2x^2 - 4x + 1, and you want to find an expression for f(u). To do that, you can just substitute
u = 2x - 3

and solve to express x in terms of u. Put this in place of x on the right side, and simplify to obtain f(u).

Another way to describe this process, if you are comfortable with the notation of function composition, is that you want $f \circ g = h$, where h(x) = 2x^2 - 4x + 1 and g(x) = 2x - 3, but you don't know f. You can just compose each side with $g^{-1}$:

$(f \circ g) \circ g^{-1} = h \circ g^{-1}$

The left side simplifies to f, so the right side is the answer.

Awesome! Found the exact same result as the "guessing" and it is much easier. Thank you!