# Solving Inverse Function Homework: Bijection & Uniqueness

• benjamin111
In summary, the conversation discusses how to show the uniqueness of the inverse of a bijection function. The idea is to prove it by contradiction, assuming there is another function that is also the inverse of the bijection function. Then, using the definition of the inverse and the surjection of the function, it can be shown that the two functions are equal.
benjamin111

## Homework Statement

My textbook states that the inverse of a bijection is also a bijection and is unique. I understand how to show that the inverse would be a bijection and intuitively I understand that it would be unique, but I'm not sure how to show that part.

## The Attempt at a Solution

My idea is to somehow say that if the inverse function is bijective and maps S -> T such that f-1(f(x))=x, then any other function that produces the same result must be the same function, but I can't quite figure out how to make this statement mathematically...
Thanks.

Well, about showing that the inverse is unique, try to prove it by using a contradiction. That is suppose that there is another function call it g that is different from f^-1 ( the inverse of f) but that is also the inverse of f, ( suppose that also g is the inverse of f) and try to derive a contradiction, in other words try to show that indeed f^-1=g.

This is the ide, the rest are details.

Suppose f:X->Y is a bijection. Let g and h be inverses of f. Show that g(y)=h(y) for all y in Y. To do this express y as f(x) (possible since f is a surjection), and use the definition of g and h.

## 1. What is an inverse function?

An inverse function is a mathematical operation that undoes another function. It switches the input and output values of the original function, such that the output of the inverse function becomes the input of the original function.

## 2. What is bijection in the context of inverse functions?

Bijection refers to a one-to-one correspondence between the elements of two sets. In the context of inverse functions, it means that each element in the domain has a unique corresponding element in the range, and vice versa.

## 3. How do I know if a function is bijective?

A function is bijective if it is both injective (one-to-one) and surjective (onto). In other words, each element in the domain is mapped to a unique element in the range, and every element in the range has a corresponding element in the domain.

## 4. Why is uniqueness important when solving inverse function homework?

Uniqueness is important because it ensures that the inverse function exists and is well-defined. If a function is not unique, it means that there are multiple possible inverse functions, which can lead to confusion and errors in solving the homework.

## 5. How do I solve a bijection and uniqueness problem for inverse functions?

To solve a bijection and uniqueness problem for inverse functions, you can use the horizontal line test to check if the function is one-to-one, and then use algebraic methods to prove that the inverse function is unique. This may involve solving for the inverse function algebraically or graphically.

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