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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.

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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.

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  • $\begingroup$ Okay so I'm working on a programming assignment that does this. I am inputting into a function all integer values from the domain of x. If I get a number back that is outside the codomain, the function is still a partial function? So it doesn't matter what I get as output? Or does at least one of the outputs have to be defined for the whole function to be a partial function? $\endgroup$ – Coder117 May 1 '17 at 1:27
  • $\begingroup$ The only valid outputs of a function are those in the codomain. If you would get an output outside that codomain, then it means you do not have a valid output for the function, therefore the function does not have an output for one of its inputs, and is therefore partial. $\endgroup$ – ConMan May 1 '17 at 1:30
  • $\begingroup$ So if all the outputs I get are not in the codomain, is the function still partial? $\endgroup$ – Coder117 May 1 '17 at 1:48
  • $\begingroup$ @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. $\endgroup$ – Bram28 May 1 '17 at 1:58
  • $\begingroup$ I understand what you are saying, but is it possible to have a function where no matter what I put in for x, I never get an output that fits within my codomain? If so, is that function a partial function? Or not at all? For example is f(x) = x + 10^12 + 1 a partial function even though when I put in my lower bound of my domain f(-10^6) I still get out 10^6 + 1 which is not in my codomain? $\endgroup$ – Coder117 May 1 '17 at 2:02

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