Prove the Function is Onto: f(x) = 1/x  Summary and Q&A
TL;DR
The video provides a stepbystep proof of the surjectivity of function f, which maps nonzero real numbers to their reciprocals.
Key Insights
 🍁 Function f maps nonzero real numbers to their reciprocals, defined as f(x) = 1/x.
 😫 To prove surjectivity, it is necessary to show that for any element B in the target set, there exists an element A in the domain set such that f(A) = B.
 💪 The proof for surjectivity involves working backwards, starting with f(A) = B and determining the value of A that satisfies the equation.
 ❓ The importance of using the correct definition of surjectivity in proofs cannot be overstated, as it ensures the validity and accuracy of the proof.
 The proof verifies that the equation f(A) = B holds true for any value B in the target set, thereby establishing the surjectivity of function f.
 😫 The function f is surjective because it covers all nonzero real numbers in its range, ensuring that every value in the target set has a corresponding preimage in the domain set.
 🚱 The proof relies on the fact that B is nonzero, allowing the reciprocal of B to be welldefined.
Transcript
so the function f and it's defined from the set of non zero reals that's what this symbol means our star into the set of nonzero reals by f of X equals 1 over X and we're going to prove that F is on  in other words F is a surjection so recall the definition of a surjective function we say that a function f from a into B is on  or subjects or surj... Read More
Questions & Answers
Q: What is the definition of a surjective function?
A surjective function is one where for every element B in the target set, there exists an element A in the domain set such that f(A) = B. In other words, every element in the target set has a preimage in the domain set.
Q: How does the proof for surjectivity of function f start?
The proof starts by considering an arbitrary value B in the target set and determining the corresponding value A in the domain set. It then proceeds to verify that f(A) indeed equals B.
Q: How is the value A determined in the proof?
The value A is determined by solving the equation f(A) = B, which is equivalent to 1/A = B. By multiplying both sides by A, we obtain 1 = BA. Dividing by B, we find that A = 1/B.
Q: Why is it important to use the correct definition of surjectivity in proofs?
Using the correct definition of surjectivity is crucial because it provides a precise criterion for proving that every element in the target set has a preimage in the domain set. Failing to use the correct definition can lead to incorrect or incomplete proofs.
Summary & Key Takeaways

Function f is defined as f(x) = 1/x, where x is a nonzero real number.

A surjective function is one where for every element in the target set, there exists an element in the domain that maps to it.

The video demonstrates a proof showing that for any value B in the target set, there exists a corresponding value A in the domain set such that f(A) = B.