Consider the following: h={(1,2),(2,4),(3,6),(-1,-2),(-2,-4)}.
a) Explain why h: ℤ --> ℤ does not describe h correctly.
I think this is probably because the range of the function h is finite or limited which means that we can't account for all possible integers. I am not sure if this is correct?
b) If the co-domain of h is the set {-4, -2, 0, 2, 4,... , 6}, is h onto? Why or why not?
I think that if the co-domain of h is the set given, then it would not be onto because the co-domain is not equal to the range of the function h. The range is missing a 0 and a -6. I'm not sure if my reasoning is correct either?
c) Find h(h(1)).
I think that h(h(1)) = h(2) = 4 ???
good answers, but B is true because there are elements of the codomain which are not used. h is into, not onto.
Thank you so much for your help! I am just wondering if my answers for part a and c are correct?
yes - that's what "good answers" means ...
I don't understand how b is considered onto?
a) Your reasoning is partially correct. The function h: ℤ --> ℤ does not describe h correctly because it does not account for all possible integers. Specifically, in the given set h={(1,2),(2,4),(3,6),(-1,-2),(-2,-4)}, there are no mappings for integers other than 1, 2, 3, -1, -2. Therefore, h: ℤ --> ℤ would only describe a subset of the integers, not all integers.
b) Your reasoning is correct. If the co-domain of h is the set {-4, -2, 0, 2, 4,... , 6}, then h is not onto. To be onto, every element in the co-domain must have a corresponding element in the domain. In this case, the co-domain has elements 0 and -6, but they do not have corresponding elements in the range of h. Therefore, h is not onto.
c) To find h(h(1)), we first calculate h(1) and then use the result as an input for h.
h(1) means finding the mapping for the input 1 in the given set h={(1,2),(2,4),(3,6),(-1,-2),(-2,-4)}. We can see that (1,2) is the mapping for input 1.
Now, using the result h(1) = 2 as the input for h, we find the mapping for input 2 in h. From the given set, (2,4) is the mapping for input 2.
Therefore, h(h(1)) = h(2) = 4.
Your answer, h(h(1)) = 4, is correct.