[tex]g \circ f(n) = \left\{ \begin{gathered} - 1,\text{ n is even } \hfill \\
+ 1, \text{ n is odd } \hfill \\ \end{gathered} \right. [/tex] gives
Any even integer squared is even. Any odd integer squared is odd.
So acts exactly as .
1. Given that f and g are both mappings of one set of integers to another set of integers, where and , determine each of the following mappings.
(1)
[Sol]
(2)
[Sol]
They are the same.
4. Given that f and g are both mappings of one set of integers to another set of integers, where and , determine each of the following mappings.
(1)
[Sol]
(2)
I'm not too sure of how that works. My understanding is that composite mapping of is you substitute whatever g is into f as its x. I don't understand this case. Any help would be appreciated.
BTW, how do I get the symbol { into latex? I understand this is used as a function in symbols, but if I actually want this symbol to appear as latex text, what do I type?
I am not too sure I understand the question, all those solutions are correct? Are they not your answers and you are looking for an explanation or what?
you just look at what g does first , then see what f does to that one.
So in the last example you noticed that squaring preserves parity. that is an even squared is even, and an odd squared is odd. Since the second function only depends on whether it is even or odd, the first function might as well just be the identity function and so you are just going to get g again.
On the second to last one, you do them in the opposite order, so for the evens you get -1 and for odds you get 1, but when you square either of these you just get 1, so this composition just always gives you 1.
Hope this helps, but I feel that you are probably asking something else entirely.
the second part of both questions is the same, like we said, any even number squared is still even, any odd number squared is still odd.
does absolutely nothing when it is applied before a function that only depends on whether the number is even or odd. That is why the second part of both questions is precisely g(x).