https://file.aiquickdraw.com/mathbot/file/1729337983851vupuzvms.jpg

The question in the image reads:

"Q5. Show that the inverse of a one-one, onto function is also one-one and onto."

This is a classic question in the study of functions, specifically about bijections. Let's break down the proof.

### Definitions:
- A **one-one function** (or injective function) ensures that no two distinct elements in the domain map to the same element in the codomain.
- An **onto function** (or surjective function) ensures that every element in the codomain has a preimage in the domain.
- A **bijective function** is both one-one (injective) and onto (surjective). Only bijective functions have inverses.

### Proof:
Let \( f : A \to B \) be a bijective function, which means it is both one-one (injective) and onto (surjective). We need to show that the inverse function \( f^{-1} : B \to A \) is also one-one and onto.

1. **Injective property of \( f^{-1} \):**
   - Suppose \( f^{-1}(y_1) = f^{-1}(y_2) \) for some \( y_1, y_2 \in B \).
   - Applying \( f \) to both sides, we get \( f(f^{-1}(y_1)) = f(f^{-1}(y_2)) \), which simplifies to \( y_1 = y_2 \), since \( f(f^{-1}(y)) = y \) for any \( y \in B \).
   - Therefore, \( f^{-1} \) is injective.

2. **Surjective property of \( f^{-1} \):**
   - Let \( x \in A \). Since \( f \) is surjective, there exists some \( y \in B \) such that \( f(x) = y \).
   - Applying \( f^{-1} \) to both sides, we get \( f^{-1}(y) = x \).
   - Thus, for every \( x \in A \), there exists a \( y \in B \) such that \( f^{-1}(y) = x \), proving that \( f^{-1} \) is surjective.

Since \( f^{-1} \) is both injective and surjective, it is also bijective. Hence, the inverse of a one-one and onto function is itself one-one and onto.

Let me know if you'd like further details or have any other questions! 

Here are 5 follow-up questions for further exploration:
1. How do you prove that a function is bijective using examples?
2. What are some real-life applications of bijective functions?
3. How can a non-bijective function be modified to have an inverse?
4. Can a function be injective but not surjective? Explain with an example.
5. What happens to the domain and codomain when taking the inverse of a bijection?

**Tip**: Remember that only bijective functions have inverses. If a function is not both one-one and onto, it does not have an inverse.

Show that the inverse of a one-one, onto function is also one-one and onto.

Learn how to prove that the inverse of a one-one, onto (bijective) function is also one-one and onto. This problem explores key concepts in functions and inverse functions, including injectivity, surjectivity, and the behavior of bijective functions. Suitable for undergraduate mathematics students or advanced high school learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation