Math question about adapting piece-wise functions to a single equation?

[Zazie J]

I am a little new to college math, so go easy on me. My question is:
Can I incorporate the piece-wise function: [(2x^2+1 , x<-1 or x>3) and (-|x| +4 , -3<x<3)] into one single function? And if so what type of math would I need to use? The values used in the function could not deviate from the original values.

I have graphed this on my calc with a piece-wise but out of pure curiousity, would like to know if a method to create this into a new function is known to anyone herein. The resulting graph would pass a vertical line test, it is also continuous, but not smooth.
I am not so sure it would be amply termed as a polynomial function anymore?
Marvin: yes I did mean that but didn't think to find use of the = with the >< signs.
Thank you guys, very helpful, it was just something I was curious about... off shoot from behavior of polynomial function of higher degree. Just fuels my fire to study harder.