3.Identify the domain, range, and codomain of {(x,y)included R x R: y=1/(cos x)}?

Question

7.give an example of a sequence x s.t. the range of x has exactly 3 elements.
8.is the following a function from the indicated domain to the indicated codomain?If it is not a function show that the function is not well defined by naming an equivalence class in the domain that is assigned two different values.
f: Z_3-->Z_6 given by f(x bar)={2x}
where we represent an element of the domain as an equivalence class x bar, and use the notation [x] for equivalence classes in the codomain.
14.For the canonical map f:Z-->Z_6, find
a)f(3)
b)the image of 6
c)a pre-image of 3 bar
d)all pre-images of 1 bar

Answer 1

3. Domain: {x ∈ R: cos x ≠ 0}
Range: (-∞, -1] ∪ [1, ∞)
Codomain: R

7. I'm not familiar with the term "range of a sequence", but it sounds like you want something like
(0, 1, 2, 0, 1, 2, 0, 1, 2, ...)

8. If (y bar) = (x bar), i.e. y ∈ (x bar) in Z_3, then y = x + 3k for some integer k. So 2y = 2x + 6k, i.e. 2y ∈ [2x] in Z_6 and so [2y] = [2x]. So this function is well defined.

14. a) f(3) = 3
b) f(6) = 0
c) By this I assume you mean a single element of what I would call the pre-image (i.e. the set of all elements mapping to 3). 3 will do, as will any number of form 3 + 6k for k in Z.
d) {1 + 6k, k ∈ Z}