Let F be a 11 and onto function with domain D and range R Pr

Let F be a 1-1 and onto function with domain D and range R. Prove the inverse function is also 1-1 and onto.

Solution

Given that f is both one-to-one and onto.

Proof for inverse function in one-to-one:

Let f is a function from a set A to a set B.

Suppose that f ^-1(y₁) = f ^-1(y₂) for some y₁ and y₂ in B.
Then since f is a onto, there are elements x₁ and x₂ in A such that y₁ = f(x₁) and y₂ = f(x₂).
Then since f ^-1(y₁) = f ^-1(y₂) by the assumption, f ^-1(f(x₁)) = f ^-1(f(x₂)) holds.
Also by the definition of inverse function, f ^-1(f(x₁)) = x₁, and f ^-1(f(x₂)) = x₂.
Hence x₁ = x₂.
Then since f is a function, f(x₁) = f(x₂), that is y₁ = y₂.
Thus we have shown that if f ^-1(y₁) = f ^-1(y₂), then y₁ = y₂.
Hence f ^-1 is an one-to-one.

Proof for inverse function in onto:

Since f is a function from A to B, for any x in A there is an element y in B such that y= f(x).
Then for that y, f ^-1(y) = f ^-1(f(x)) = x, since f ^-1 is the inverse of f.
Hence for any x in A there is an element y in B such that f ^-1(y) = x.
Hence f ^-1 is a onto.

Hence f ^-1 is both one-to-one and onto.