Problem 62 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Prove that the function \(y=\frac{k}{x}\) is inverse to itself.

Step-by-Step Solution

Answer

The inverse of the given function \(y = \frac{k}{x}\) is found by swapping x and y, resulting in \(x = \frac{k}{y}\). Solving for y, we find that the inverse function is also \(y = \frac{k}{x}\). Since the inverse function is the same as the original function, we can conclude that the function is inverse to itself.

Step 1: Write down the given function

The given function is \(y = \frac{k}{x}\).

Step 2: Swap x and y

To find the inverse of the function, we need to switch the roles of x and y. This means we will swap x and y in the given equation and get: \(x = \frac{k}{y}\).

Step 3: Solve for y

Now, we need to solve the equation for y. To do this, we can rearrange the equation in the following way: \(xy = k\) Now, divide both sides by x to isolate y: \(y = \frac{k}{x}\)

Step 4: Verify that the inverse is the same as the original function

Since the inverse function is also \(y = \frac{k}{x}\), we can conclude that the original function is inverse to itself.

Key Concepts

Function Inverses

Understanding function inverses is crucial in mathematics as they provide a way to 'undo' the action of a function. Think of it as the mathematical equivalent of a reverse gear in a vehicle. Given a function that takes you from point A to point B, the inverse function is the one that can take you back from point B to point A.

The process of finding an inverse involves swapping the input (usually represented by x) and the output (usually represented by y), and then solving for the new output variable. In our exercise, we demonstrate this by replacing x with y and vice versa, leading to the conclusion that the given function is its own inverse. This intriguing property is not common among functions and indicates a deep symmetry in the behavior of the function represented by the equation \(y = \frac{k}{x}\).

Algebraic Manipulation

Algebraic manipulation is akin to a toolset that enables you to rearrange, simplify, or solve mathematical equations. It involves a variety of techniques such as adding, subtracting, multiplying, dividing both sides of an equation, factoring, expanding, and many others. In the context of our exercise, algebraic manipulation comes into play when we isolate y in the equation \(x = \frac{k}{y}\).

By multiplying both sides of this equation by y and then dividing both sides by x, we smoothly transition to the form \(y = \frac{k}{x}\), showcasing a direct use of algebraic manipulation. Becoming proficient in these techniques is not only essential for finding function inverses but is also a fundamental skill set for solving a broad spectrum of problems in algebra.

Function Properties

Every function comes with a set of inherent properties that define its behavior and characteristics. Some key properties include the domain and range, the linearity or non-linearity, symmetry, asymptotes, and whether a function is one-to-one or many-to-one — which directly impacts the existence of an inverse function.

In our example, the function \(y = \frac{k}{x}\) exhibits a property known as self-inverseness. This means that the function acts as its own inverse. One prerequisite for a function to have an inverse is that it must be a one-to-one function, meaning each input is mapped to a unique output. However, our function challenges this standard view because it displays symmetry about the identity line, \(y = x\), which is a characteristic graphical representation of self-inverse functions. Its domain and range, which exclude zero, also support the existence of the inverse.