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.