# Thread: Compositum of the functions

1. ## Compositum of the functions

We are given $\displaystyle f(x) = \dfrac{3x -1}{x +1}$ and $\displaystyle g(x) = 2x + 1$. We need to find function $\displaystyle h(x)$, if this is true: $\displaystyle f(g(x)) = g(h(x))$. How can I find this function $\displaystyle h(x)$?

I figure it out that $\displaystyle f(g(x)) = \dfrac{6x + 2}{2x + 2}$, but I don't know how it goes next step?

2. ## Re: Compositum of the functions

Step 1: calculate $f(g(x))$. Step 2: set $g(h(x))=f(g(x))$. Step 3: solve for $h(x)$

Where are you struggling.

3. ## Re: Compositum of the functions

How can I solve for $\displaystyle h(x)$?

4. ## Re: Compositum of the functions

$$g(h(x)) = \dfrac{6x+2}{2x+2}$$

$$2h(x)+1 = \dfrac{6x+2}{2x+2}$$

$$h(x) = \dfrac{1}{2}\left(\dfrac{6x+2}{2x+2}-1\right) = \dfrac{x}{x+1}$$

5. ## Re: Compositum of the functions

Ok, thanks for your help! Woow, why I couldn't figure it out at first place... :P Thank you again!