Question

Do all functions have inverses

Answer 1

Not all functions have inverses. For a function to have an inverse, it must be both one-to-one (injective) and onto (surjective). Functions failing to meet these criteria, such as
`\(f(x) = x^2\)`, lack inverses.

No, not all functions have inverses. For a function to have an inverse, it must satisfy certain conditions:

1. **One-to-One (Injective):** The function must be one-to-one, meaning that no two different inputs map to the same output. In other words, each distinct input corresponds to a unique output.

2. **Onto (Surjective):** The function must be onto, meaning that every element in the codomain (the set of possible outputs) is covered by the function. In simpler terms, the function's range must be the entire codomain.

If a function meets these criteria, it is said to be bijective, and it has an inverse function. However, many functions do not meet these conditions. For example, the function
`\(f(x) = x^2\)` is not one-to-one because different inputs can yield the same output (e.g., f(-2) = f(2) = 4), so it does not have an inverse.