# Functions Problem 4

Recap of definitions:
\(h:\mathbb{Z}\rightarrow \mathbb{Z}\) is known to be onto.
\(f:\mathbb{Z}^2\rightarrow \mathbb{Z}\) is defined
by \(f(x,y) = h(x)+h(y)\).

### Getting started

Before trying to write the proof, you need to figure out how
to take an output value k and find a pre-image for it.
For example, suppose k=17. How are you going to find values
for x and y such that f(x,y) = 17?

Notice that h is known to be onto. So if I pick some value of k,
e.g. k=17, I know there is an x such that h(x)=17. I don't know
exactly what x is, but I know it exists. This also works for
special values. E.g. if I want to have h(x) = 1, there is a value
for x that will do that.

It might help to first consider the simpler function g, defined
by g(x,y) = x + y. How would you find input values for x and y
that would give you (say) 17 as the output value? If I give you
a symbolic output value k, can you find me values for x and y
such that g(x,y) = k?

### Writing the final draft

Once you've solved the above, you need to write out your proof
in logical order. Pick a random output value k. Then tell the reader
your choices for x and y (typically defined in terms of k and h).
Then verify that f(x,y) = k.