Let f : Z → Z be such that f(x) = x + 1. Is f invertible, and if it is, what is it’s inverse?

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To determine if the function is invertible, we prove that it is a bijection, i.e it is both one-to-one and onto

Injective/ one-to-one

If f(x)=f(y), then x=y i.e

x+1=y+1
x=y+1-1
x=y
Thus f(x) is injective

Surjective / onto

f(x)=x+1, but x=y-1
f(x)=(y-1)+1
f(x)=y
Thus f(x) is surjective.

f(x) is bijective since it is both injective and surjective, hence it is invertible

Inverse

y=x+1
x=y-1
Thus the inverse of f(x) is given by:
f¯¹(y)=y-1 or g(y)=y-1