Problem 13 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

A condition for a function \(y=f(x)\) to have inverse is that it should be (a) defined for all \(x\) (b) continuous everywhere (c) an even function (d) strictly monotonic and continuous in the domain

Step-by-Step Solution

Answer

Option (d): strictly monotonic and continuous in the domain.

Step 1: Understanding Inverses

For a function to have an inverse, it must be bijective, meaning both injective (one-to-one) and surjective (onto). This ensures that each input maps to a unique output and vice versa.

Step 2: Identifying Key Properties

For a function to be injective, it must be strictly monotonic within its domain. This means that the function should be either entirely non-decreasing or non-increasing, ensuring each value in the range is attained exactly once.

Step 3: Considering Continuity

Additionally, the function should be continuous in its domain to ensure the inverse function is also continuous. Continuity helps maintain a single, well-defined output for each input.

Step 4: Asses the Provided Options

Given the options: (a) Defined for all x - This doesn’t ensure the function is one-to-one. (b) Continuous everywhere - Continuity alone isn’t enough for inverses. (c) Even function - An even function is not one-to-one. (d) Strictly monotonic and continuous in the domain - This ensures a one-to-one mapping required for an inverse.

Key Concepts

Bijection

A bijective function is one that is both injective and surjective. In simple terms, every output in the function's range must be linked to exactly one input. This property is essential for a function to have an inverse.
Each input in the domain should map to a unique output, and every possible output in the range should have a corresponding input.
Without these conditions, some values could map to the same output, or some outputs might have no corresponding inputs. Either of these issues would prevent a function from having a well-defined inverse.

Injective Functions

An injective function is also known as a one-to-one function. This means that each input produces a unique output that no other input produces.
Consider a scenario where multiple inputs produce the same output. Here, it would be impossible to reverse the function because, given the output, you would not know which input to attribute it to.

• Must have distinct outputs for distinct inputs
• Aids in guaranteeing that the function is a bijection

Ensuring injectiveness is often aided by the property of monotonicity in a function.

Monotonicity

Monotonicity describes the behavior of a function in terms of it being consistently increasing or decreasing. This property helps in achieving injectiveness.

• If a function is always increasing, it means each increase in input results in a unique increase in output.
• If a function is consistently decreasing, each decrease in input results in a unique output.

Monotonicity ensures that no two different inputs can lead to the same output, which is crucial for a function to be invertible.

Continuity

Continuity in a function means it has no abrupt changes, jumps, or breaks. This property ensures that small changes in the input produce small changes in the output.

• Allows for smooth transition and mapping across the domain
• Ensures the inverse will also display predictable behavior

In terms of having an inverse, continuity is crucial. A continuous function will have an inverse that is defined and smooth, preventing gaps or overlaps that could disrupt the one-to-one correspondence necessary for inversion.