How do we decompose functions with fraction powers

Question

Here is an example

$\dfrac{1}{2+\sqrt x}$

I am tempted to use the form $\dfrac{Ax+C}{2+\sqrt x}$

Which will give me 1=Ax+C

And if x=0 then C= 1, A= 0

This brings us to the same function we started with, indicating that this is not decomposable? right?

Answer 1

The only thing I can assume you are trying to do is solve an integral..that being said what you are trying to do is probably not what you want to do (correct me if I am wrong?).

$$ \frac{1}{2+\sqrt{x}} = \frac{2-\sqrt{x}}{(2+\sqrt{x})(2-\sqrt{x})} = \frac{2-\sqrt{x}}{4-x} $$ thus you could right the above as $$ \frac{1}{2+\sqrt{x}} = \frac{2}{4-x} -\frac{\sqrt{x}}{4-x} $$ beyond that, I am clueless.

or define $u = f(x)$ which also may not be what you want to achieve.

Answer 2

Decomposing this to the most basic functions would give:

$$ f(x)=\frac{1}{x},\qquad g(x)=2+x, \qquad h(x)=\sqrt x $$

So that $f(g(h(x)))=\frac{1}{2+\sqrt x}$. Were you referring to some other form of decomposition?

Answer 3

You are correct, this is not decomposable.