How to simplify / combine function equtions containing `if`? I have two eqution (blue / green), composed them using a simple if (black).

Blue: (2x)^2 / 2
Green: 0.5 + (1 - (2(1-x))^2) / 2
Black: (x < 0.5) ? (2x)^2 / 2 : 0.5 + (1 - (2(1-x))^2) / 2
See it in action on Fooplot: http://fooplot.com/plot/7knnh3zxis

Question is how can I simplify / extract the if statement? 
Seemingly this is a single eqution offset by 0.5, 0.5 from the origin. Can someone shed a light on how to simplify this? I have some other similar equtions I am to simplify.
I'm not a math ninja, so explain with care if you can. :)
Sorry if this is a noob question here at math.stackexchange.com, this is the first time I ask here.

UPDATE: I expressed as a single function, than it turned out that this all above can be simplified to a surprisingly simple function, like:
-(4(x-0.5)abs((x-0.5))-4(x-0.5)-1)/2
See http://fooplot.com/plot/t5msqna67s
 A: You can't write the whole function you have defined as a polynomial (sum of multiples of $x^n$). Your function is differentiable everywhere and piecewise a polynomial, but that doesn't mean that the function as a whole is a polynomial. To see this, note that your function's second derivative is discontinuous.
Although it's kind of cheating, you could write it as:
$$f(x):=\frac{2x^2}{2}+ U\left(x-\frac{1}{2}\right)\cdot \left[\frac{1}{2} + \frac{1 - (2(1-x))^2}{2} - \frac{2x^2}{2}\right]$$
with 'U' being the 'unit step': $U(x)=\left\{\begin{array}{ll}0 & x<0 \\ 1 & x \geq 0\end{array}\right.$
Note that some polynomials of degree 3 will have a similar shape to your function, but they will go off to infinity faster on their tail ends.
A: If you with extracting mean to replace the "if" with some function then you could use the following function:
$\displaystyle t(x)=\frac{x+|x|}{2\cdot x}$ which is equal to
$ \;t(x)=\!\!
\left\{ 
\begin{array}{l}
1 \quad\text{if } x>0\\ 
0 \quad\text{if } x<0
\end{array}
\right. 
$
In your example $f(x)=\text{blue}(x)\cdot t(x-0.5)+\text{green}(x)\cdot t(0.5-x)$.
