Question

Let $ f : x \to y $ be a given function, then $ f^{-1} $ exists if

• A. \( f \) is one-one
• B. \( f \) is onto
• C. \( f \) is one-one but not onto
• D. \( f \) is one-one and onto

Hint:

For a function to have an inverse, it must be bijective, which means it must be both one-to-one (injective) and onto (surjective).

Answer 1

The Correct Option is D

Step 1: Understand the condition for the inverse function.
For a function \( f \) to have an inverse, it must be both one-to-one (injective) and onto (surjective).
A one-to-one function ensures that every element of the domain maps to a unique element in the codomain, and an onto function ensures that every element in the codomain has a corresponding element in the domain.
Step 2: Analyze the options.
Option (a): \( f \) is one-one: A one-to-one function is necessary for the existence of the inverse, but it is not sufficient alone.
The function also needs to be onto. Option (b): \( f \) is onto: Being onto is necessary but not sufficient.
A function must be both one-to-one and onto for an inverse to exist. Option (c): \( f \) is one-one but not onto: This is not enough for the inverse to exist because the function is not onto. Option (d): \( f \) is one-one and onto: This is the correct condition.
A function that is both one-to-one and onto is bijective, and only bijective functions have inverses.
Step 3: Conclusion.
For an inverse function to exist, the function must be bijective (both one-to-one and onto).