Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
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.
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 $f(x) = 4x^2.$
