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

  • $\begingroup$ What does $\sqrt Z$ mean above? Also, what is $X$? $\endgroup$ – Teresa Lisbon Feb 15 '19 at 14:52
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    $\begingroup$ 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. $\endgroup$ – 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 $f(x) = 4x^2.$


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