What condition makes a function g: R → R invertible?

What condition makes a function g: R → R invertible?

• A. The function must be surjective
• B. The function must be injective (one-to-one)
• C. The function must be continuous
• D. The function must be differentiable

Answer: (B)

The main idea is that an inverse function exists only when each output comes from a unique input. That one-to-one (injective) property guarantees that you can unambiguously recover the original input from the output. If a function assigns the same value to two different inputs, there isn’t a single preimage for that output, so you can’t define a single inverse value as a function. For example, x^2 maps both 1 and -1 to 1, so it cannot have a proper inverse on all of R. On the other hand, a function like x^3 assigns each real number to a unique real number, so it has an inverse. Continuity or differentiability don’t ensure invertibility, because they don’t prevent multiple inputs from producing the same output. Surjectivity would guarantee that every real number is hit, but even then you’d still need injectivity to have a well-defined inverse function.

The main idea is that an inverse function exists only when each output comes from a unique input. That one-to-one (injective) property guarantees that you can unambiguously recover the original input from the output.

If a function assigns the same value to two different inputs, there isn’t a single preimage for that output, so you can’t define a single inverse value as a function. For example, x^2 maps both 1 and -1 to 1, so it cannot have a proper inverse on all of R. On the other hand, a function like x^3 assigns each real number to a unique real number, so it has an inverse.

Continuity or differentiability don’t ensure invertibility, because they don’t prevent multiple inputs from producing the same output. Surjectivity would guarantee that every real number is hit, but even then you’d still need injectivity to have a well-defined inverse function.