If a function is one-to-one, what can be said about its inverse?

If a function is one-to-one, what can be said about its inverse?

• A. The inverse is a function.
• B. The inverse is not a function.
• C. The inverse exists only if the domain is finite.
• D. The inverse is always linear.

Answer: (A)

When a function is one-to-one, each output value comes from exactly one input value. That means you can swap the roles of inputs and outputs without ambiguity: for every y in the function’s image, there is a unique x with f(x) = y, so the inverse can send that y back to that x. The inverse, then, is a function whose domain is the original function’s range and whose codomain is the original domain. It doesnures on finite size; no need to be onto for the inverse to exist on its natural domain. For example, if f(1)=2, f(2)=3, and f(3)=5, then the inverse maps 2→1, 3→2, and 5→3. Therefore, the inverse is a function.

When a function is one-to-one, each output value comes from exactly one input value. That means you can swap the roles of inputs and outputs without ambiguity: for every y in the function’s image, there is a unique x with f(x) = y, so the inverse can send that y back to that x. The inverse, then, is a function whose domain is the original function’s range and whose codomain is the original domain. It doesnures on finite size; no need to be onto for the inverse to exist on its natural domain. For example, if f(1)=2, f(2)=3, and f(3)=5, then the inverse maps 2→1, 3→2, and 5→3. Therefore, the inverse is a function.