Problem 100 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f: N \rightarrow N\), where \(f(x)=x+(-1)^{x-1}\). Then, (A) \(f^{-1}(x)=x+(-1)^{x-1}\) (B) \(f^{-1}(x)=x\) (C) \(f^{-1}(x)=x-(-1)^{x-1}\) (D) None of these

Step-by-Step Solution

Answer

(C) \(f^{-1}(x) = x - (-1)^{x-1}\).

Step 1: Understanding the Function

We have the function \(f: \mathbb{N} \rightarrow \mathbb{N}\) where \(f(x) = x + (-1)^{x-1}\). We need to find its inverse \(f^{-1}(x)\). The expression \((-1)^{x-1}\) will be -1 if \(x\) is even and 1 if \(x\) is odd. Therefore, for even \(x\), \(f(x) = x - 1\), and for odd \(x\), \(f(x) = x + 1\).

Step 2: Solving for the Inverse

To find the inverse, we analyze two cases separately: 1. **Odd inputs:** If \(f(x) = x + 1\), then to solve for \(x\) given \(f(x) = y\), we have \(y = x + 1\). Thus, \(x = y - 1\).2. **Even inputs:** If \(f(x) = x - 1\), then to solve for \(x\) given \(f(x) = y\), we have \(y = x - 1\). Thus, \(x = y + 1\).

Step 3: Determining the Inverse Function

From both cases, the inverse function based on whether \(x\) is odd or even is given by: - If \(x\) is obtained from an odd number \(y\) (i.e., \(y = x + 1\)), then to get back to \(x\), we subtract 1: \(f^{-1}(y) = y - 1\).- If \(x\) is obtained from an even number \(y\) (i.e., \(y = x - 1\)), then we add 1 to get back to \(x\): \(f^{-1}(y) = y + 1\).Therefore, \(f^{-1}(x) = x - (-1)^{x-1}\), which simplifies to \(f^{-1}(x) = x - (-1)^{x-1}\).

Step 4: Matching with Given Options

Among the given options, option (C) \(f^{-1}(x) = x - (-1)^{x-1}\) matches our derived inverse function. This means option (C) is correct.

Key Concepts

Functions in Mathematics

In mathematics, functions are fundamental concepts that establish relationships between inputs and outputs. A function is like a machine that takes an input, performs a specific operation, and produces an output. Formally, we denote a function as \( f: A \to B \), where \( A \) is the domain (all possible inputs), and \( B \) is the range (all possible outputs).

Functions are defined by rules that apply to the elements of their domain. Each input must have exactly one output, making functions predictable and reliable. A simple example is the function \( f(x) = x^2 \), mapping each input \( x \) to its square.

Understanding functions is crucial as they model real-world scenarios, allowing us to analyze and solve problems in various fields like physics, engineering, and economics. Many different types of functions exist, including linear, quadratic, exponential, and trigonometric functions.

Odd and Even Numbers

Odd and even numbers are a basic concept in mathematics, defining all integers. An even number is one that can be divided by 2 without leaving a remainder, such as 2, 4, 6, etc. Conversely, odd numbers do not evenly divide by 2, examples being 1, 3, 5, and so on.

These numbers have distinct properties and consequences. For instance, the sum or difference of two even numbers is always even, and the sum of an odd and an even number is always odd. In this exercise, understanding whether \( x \) is odd or even directly affects the value of \( (-1)^{x-1} \):

• If \( x \) is even, \((-1)^{x-1} = -1\), making it essential for calculations.
• If \( x \) is odd, \((-1)^{x-1} = 1\), altering the equation.

Knowing the parity (oddness or evenness) of a number can simplify and solve problems involving integers, like determining function outputs or inverses.

Function Inversion

Inversion of a function is a process of finding another function that "reverses" the effect of the original one. If you have a function \( f(x) \) mapping input \( x \) to output \( y \), then the inverse function \( f^{-1}(y) \) works backwards: it takes \( y \) and returns \( x \).

To find an inverse function, you often solve the equation \( y = f(x) \) for \( x \). Once rearranged, you swap \( x \) and \( y \) to find \( f^{-1}(x) \).

The original exercise involved calculating \( f^{-1}(x) \) for a function based on if \( x \) was odd or even. This involved:

• For odd \( x \), the inverse formula is \( f^{-1}(x) = x - 1 \).
• For even \( x \), the inverse becomes \( f^{-1}(x) = x + 1 \).

Thus, the correct inverse was identified, aligning with option \( C: f^{-1}(x) = x - (-1)^{x-1} \). Inverses are key in verifying transformations and processes in math, ensuring you can return to any initial condition.