Every bijective function has an inverse.

Solution

Maths sir · 4 steps

To reverse a function, every output must come from exactly one input, and bijective gives you exactly that.

1. What an inverse needs. An inverse f⁻¹: B → A must send each y in B back to exactly one x in A. So every y needs at least one preimage, and no y can have two.
2. Onto and one-one. Onto means every y in B has at least one preimage, so f⁻¹(y) always exists. One-one means no two inputs share an output, so f⁻¹(y) is never ambiguous.
3. An example. f(x) = 2x + 3 on ℝ is one-one and onto. Solving y = 2x + 3 gives the single value x = (y-3)/2, so f⁻¹(y) = (y-3)/2.
4. So. A bijective function meets both conditions, so its inverse always exists. The statement is true.

The idea to keep. A function is invertible exactly when it is bijective. One-one stops an output from going back to two inputs, and onto makes sure every output has somewhere to go back to.

Answer from the NTA final answer key. Solution by PrepWiser.