Showing that a given function is one-to-one, and finding its inverse

Question

Given the function f: ℤ→ℤ, defined by f(x)=x - 1

i. How can I show that the function f, is on-to-one, and onto f: ℤ→ℤ, defined by f(x)=x-1

ii. How can I find the inverse, f^-1(x) and state its domain and range

Answer 1

To show that f is one-to-one, assume that f(a) = f(b) and then show that a = b:

Suppose that a, b ε Z and f(a) = f(b).

SInce f(a) = f(b), a - 1 = b - 1. So, a = b.

Therefore, f is one-to-one.

To show that f is onto, let a ε Z. Find b ε Z so that f(b) = a:

Let a ε Z. Then b = a + 1 ε Z and f(b) = (a + 1) - 1 = a.

Therefore, f is onto.

Domain f^-1(x) = Range f(x) = Z (since f is onto)

Range f^-1(x) = Domain f(x) = Z

Answer 2

If the function has an inverse, it is one-to-one and onto.

To find the inverse, interchange the roles of x and y and solve for y.

The inverse is y=x+1.

Show that it is the inverse by substituting the expression for f inverse into f .