Let f Z Z be defined by fx 2x 7 a Is f Z Z injective one

Let f : Z Z be defined by f(x) = 2x + 7.

(a) Is f : Z Z injective (one-to-one)? Prove your answer.

(b) Is f : Z Z surjective (onto)? Prove your answer.

(c) Is f : Z Z a bijection? Prove your answer.

Solution

a.) f(x) = 2x+7. Let y = 2x+7 this implies x = (y-7)/2 which is inverse of f(x). So we can easily see that for each unique value of x there exists a unique value of y and hence Onte-to-One.
b.) as we could see from the inverse of f(x) that for each value of y in Z we can not find x in Z, and hence not Onto.

c.) f(x) : Z->Z is one to one but not onto so it is not bijection.