Proof: Invertibility implies a unique solution to f(x)=y | Linear Algebra | Khan Academy

TL;DR

Invertible functions have a unique solution for every value in the co-domain.

Transcript

I've got a function f and it's a mapping from the set x to the set y. And let's just say for the sake of argument, let's say that f is invertible. What I want to know is what does this imply about this equation right here. The equation f of x is equal to y. I want to know that for every y that's a member of our co-domains. So for every y, let me wr... Read More

Key Insights

• 🥶 Invertible functions have unique solutions for every value in the co-domain.
• 🤠 Invertibility is determined by the existence of a unique solution to f(x) = y for every y in the co-domain.
• ❓ Invertibility implies the existence of an inverse mapping function.
• ☺️ The composition of the inverse function with the original function results in the identity function on x, and vice versa for the composition on y.

Questions & Answers

Q: What does it mean for a function to be invertible?

A function is invertible if it has a unique solution for every value in the co-domain. It means that each element in the set x maps to a unique element in the set y.

Q: How can we determine if a function is invertible?

One way is to check if for every y in the co-domain, there exists a unique solution x that satisfies f(x) = y. If this condition is met, the function is invertible.

Q: What is the significance of invertibility in functions?

Invertibility ensures that there is a one-to-one correspondence between elements in the set x and the set y. It allows us to find unique solutions to equations and perform operations such as finding the inverse function.

Q: How does invertibility relate to the uniqueness of solutions?

Invertibility guarantees that for every y in the co-domain, there is a unique solution x that satisfies f(x) = y. This uniqueness is crucial in ensuring that the function can be inverted and has a well-defined inverse mapping.

Summary & Key Takeaways

• Invertible functions have a defined relationship between the set x and set y, with each element in x mapping to a unique element in y.

• If an invertible function f maps a point a to f(a), and there is a unique solution x that satisfies f(x) = b, then the function is invertible.

• Conversely, if for every y in the co-domain there is a unique solution x that satisfies f(x) = y, then the function is invertible.

• Invertibility implies the existence of a mapping function f inverse, such that the composition of f inverse with f is equal to the identity function on x, and the composition of f with f inverse is equal to the identity function on y.