# Determine the domain, co-domain and range - again

I am trying to determine the domain, co-domain and range of the following function

A function assigns to each integer, the square of that integer multiplied on 4.

What I think it is, is: f: X→√ℤ*4

But I suspect that I have some of the symbols in the wrong place?

I previously posted a similar question:

A function assigns to each bit string, the number of zeroes in that bit string.

And got a really good answer here

• What does $\sqrt Z$ mean above? Also, what is $X$? – Teresa Lisbon Feb 15 '19 at 14:52
• Based on that previous answer, what do you think the domain of this function might be ("a function assigns to each integer ..." - that's a big clue) ? What do you think the range might be ? Don't worry about symbols for now, just say what you think the answers might be in words. Co-domain requires a bit more thought because in this case not everything in the range is in the co-domain. – gandalf61 Feb 15 '19 at 14:56

Like last time, take things one step at a time. For the domain we're told that $$f$$ acts on the integers, so the domain is going to be all of $$\mathbb{Z}$$. We're told that $$f$$ acts by squaring its argument and multiplying by $$4$$, so we know that $$f(x)=4x^2$$. Since any integer multiplied by an integer yields another integer, we can see that the co-domain must be some subset of the integers again. But it's not all of the integers, because squaring an integer always gives us a positive integer. We can write that as $${\mathbb Z}^+$$ or $${\mathbb N}$$ -- in this case, I'll choose the first to emphasize that this is a subset of the integers. So $$f:{\mathbb Z} \rightarrow {\mathbb Z}^+$$ Finally, the range is only a subset of the positive integers -- for example, there is no $$x$$ such that $$f(x) = 3$$. In fact, only positive integer multiples of $$4$$ can be in the range of $$f$$ because $$f(x)=4x^2$$. So the range is a set: $$\mbox{Ran}(f) = \{ x\in {\mathbb Z}^+ : 4|x \mbox{ and } x \mbox{ is square} \}$$
I think you are principally confused with notation. The notation $$f:X\rightarrow Y$$ just specifies that $$f$$ is a function from its domain $$X$$ to its codomain $$Y$$ - it doesn't specify how the function is defined.
E.g. Let $$g:\mathbb{Z}\rightarrow\mathbb{Z}$$. We now need to specify how $$g$$ maps an integer to an integer. An example would be $$g: x\mapsto 2x$$ which is equivalent to the notation $$g(x) = 2x$$ where we double the input.
In your case you are told that your function $$f$$ maps from the integers and are given the rule, namely $$f(x) = 4x^2.$$