Partial functions

I'm still confused on what a partial function is.

If I have a domain and codomain of integers from $-10^6$ to $10^6$, what are some partial functions that work with this domain and codomain? What are some that don't?

I know for example that $x^2$ is not a function within these restrictions because if I put in $10^6$, I get $10^{12}$ out, which is out of my interval.

If someone could describe in layman's terms some partial functions and what makes them partial that would be very helpful.

The fact that the square of $10^6$ lays outside the co-domain actually makes $x^2$ a fine example of a partial function: apparently we cannot define the function value for all of the objects in the domain, and thus the function is a partial function.
For an alternative example, take $f(x)=x+1$ which will be defined for all numbers in the domain except for $10^6$, for which the function is undefined. This function is therefore also a partial function.
• @Coder117 You never get an output that is not in the codomain. So, for my example of $f(x)=x+1$, $f(10^6)$ is not defined, i.e. There is no output in that case ... Which is exactly what makes it a partial function. – Bram28 May 1 '17 at 1:58