Prove the Function is Onto: f(x) = 1/x

TL;DR

The video provides a step-by-step proof of the surjectivity of function f, which maps non-zero real numbers to their reciprocals.

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

Key Insights

• 🍁 Function f maps non-zero 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 non-zero real numbers in its range, ensuring that every value in the target set has a corresponding pre-image in the domain set.
• 🚱 The proof relies on the fact that B is non-zero, allowing the reciprocal of B to be well-defined.

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 pre-image 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 pre-image 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 non-zero 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.