3 Answers

Adding to applebottomjean's answer, when you multiply ANY number by an EVEN number (2, 4, 6, 8, 10, etc), the answer will BE an even number.  Adding 1 to that will ALWAYS be an ODD number. 
Just a little more explanation....... 

It'll be odd, because of this:

x^2, whether it's even or odd is always gonna be divisible by 2, because the function is 2x^2

So, because you're adding '1' to this EVEN number, the answer MUST BE ODD :')

Hope this helps!

Regarding even or odd function, we have following definition.

Even function : The function is even if it satisfy the relation: f(x) = f(-x) 

For example: f(x) = x^2 is even function since f(x) =f(-x) = x^2

Another example is : f(x) = cosx 

 Odd function: The function is odd if it satisfy the relation : f(x) = -f(-x)

For example: f(x) = sinx

f(x) = x^3, etc

For the given problem , f(x) = 2x^2 +1 which satisfy the relation f(x) = f(-x) 

So, it is an even function. 

Hope this helps you to clear your doubt about even or odd function.