Final answer:
The function M(x)=(2x+5) mod 7 is a function because it gives a single output for each input in Z. It must be tested for being one-to-one by seeing if different inputs yield different outputs and for being onto by checking all possible outputs mod 7 are achievable.
Step-by-step explanation:
To show that M: Z→Z defined by M(x)=(2x+5) mod 7 is a function, we need to verify that for each input x there is exactly one output. This is true for modular arithmetic functions, as each input value will yield exactly one result after applying the modulus operation. Therefore, M is indeed a function.
To determine if M is one-to-one, we must ensure that every unique input x produces a unique output M(x). We can test this by calculating the outputs for each x value in the set {0, 1, 2, 3, 4, 5, 6} which represent all possible remainders mod 7. If two different inputs give the same output, M is not one-to-one.
To verify if M is onto, we need to show that for every y in Z, there is an x in Z such that M(x) = y mod 7. With the finite set of remainders {0, 1, 2, 3, 4, 5, 6}, we can demonstrate whether each is achievable by M(x) for some value of x.