Problem 162 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f: N \rightarrow Y\) be a function defined as \(f(x)=4 x+3\), where \(Y=\\{y \in N: y=4 x+3\) for some \(x \in N\\}\). Show that \(f\) is invertible and its inverse is (A) \(g(y)=\frac{3 y+4}{3}\) (B) \(g(y)=4+\frac{y+3}{4}\) (C) \(g(y)=\frac{y+3}{4}\) (D) \(g(y)=\frac{y-3}{4}\)

Step-by-Step Solution

Answer

The inverse function is (D) \(g(y) = \frac{y-3}{4}\).

Step 1: Understanding the Function

We are given a function \( f: \mathbb{N} \rightarrow Y \) where \( f(x) = 4x + 3 \). This means for each natural number \( x \), \( f \) assigns the number \( 4x + 3 \) in the set \( Y \). We need to determine if the function is invertible and find the correct inverse.

Step 2: Condition for Invertibility

A function is invertible if and only if it is bijective (one-to-one and onto). Since \( f \) is defined as a linear function with integer coefficients, it is naturally one-to-one because each distinct \( x \) gives a distinct \( 4x + 3 \). Hence, \( f \) is injective. To show \( f \) is surjective onto set \( Y \), we note any \( y \) in \( Y \) has the form \( 4x + 3 \), ensuring every element in \( Y \) is reached by some \( x \). Thus, \( f \) is surjective.

Step 3: Finding the Inverse Function

To find the inverse, assume \( y = f(x) = 4x + 3 \). To solve for \( x \) in terms of \( y \), we rearrange the equation: \( y - 3 = 4x \). Consequently, \( x = \frac{y - 3}{4} \). This function \( g(y) = \frac{y - 3}{4} \) is proposed as the inverse.

Step 4: Verification of the Inverse

To verify, we check if \( f(g(y)) = y \) and \( g(f(x)) = x \). Firstly, substituting \( g(y) \) into \( f \), we get: \( f(g(y)) = f\left(\frac{y - 3}{4}\right) = 4\left(\frac{y - 3}{4}\right) + 3 = y - 3 + 3 = y \). Now, substituting \( f(x) \) into \( g \), so \( g(f(x)) = g(4x + 3) = \frac{(4x + 3) - 3}{4} = 4x/4 = x \). Both compositions accurately return their inputs, confirming \( g(y) = \frac{y - 3}{4} \) is the correct inverse.

Key Concepts

Injective Function

An injective function, also known as a one-to-one function, is a type of function where each element of the domain is mapped to a unique element in the codomain. This means that different inputs lead to different outputs. In mathematical terms, a function \( f: A \rightarrow B \) is injective if, whenever \( f(a_1) = f(a_2) \), it must follow that \( a_1 = a_2 \).
For our function \( f(x) = 4x + 3 \), it is injective because the expression \( 4x + 3 \) uniquely determines the output for every natural number \( x \).
To determine injectivity, we can use a simple example or test by setting \( f(x_1) = f(x_2) \) and showing that this implies \( x_1 = x_2 \).
Let's illustrate this with some intuition:

• If \( f(1) = 7 \) and \( f(2) = 11 \), you can see there's no overlap in value, meaning each input (1 and 2) results in a distinct output (7 and 11).
• Since no two different \( x \) values produce the same \( f(x) \) value, the function is confirmed injective.

Surjective Function

A surjective function, or onto function, ensures that every element of the codomain is mapped from the domain. In other words, every possible value in the output set is achieved by some input.
For a function \( f: A \rightarrow B \) to be surjective, for every \( b \) in \( B \), there exists at least one \( a \) in \( A \) such that \( f(a) = b \).
In our example, the function \( f(x) = 4x + 3 \) is surjective when considered over its restricted codomain \( Y \), where \( Y = \{y \in \mathbb{N} : y = 4x + 3 \text{ for some } x \in \mathbb{N}\} \). This essentially means every number in \( Y \) can be expressed in the form \( 4x + 3 \) for some natural number \( x \).
Let's illustrate with another approach:

• Imagine a target set \( Y \), constructed specifically such that each \( y \) in \( Y \) can be written as \( 4x + 3 \).
• Then, by construction, any \( y \) you pick from \( Y \) must correspond to some input \( x \) naturally.

This guarantees the function doesn't leave any element in \( Y \) unmapped, affirming its surjectivity.

Function Inverse

The concept of a function inverse is pivotal when discussing invertible functions. A function \( f: A \rightarrow B \) is invertible if there exists a function \( g: B \rightarrow A \) such that:

• \( f(g(y)) = y \) for every \( y \) in \( B \)
• \( g(f(x)) = x \) for every \( x \) in \( A \)

These conditions ensure that applying \( f \) and then \( g \) (or vice versa) brings you back to where you started, like seamlessly untying a knot.
For our given function \( f(x) = 4x + 3 \), finding its inverse requires solving for \( x \) in terms of \( y \). If we have \( y = 4x + 3 \), rearranging gives us \( x = \frac{y - 3}{4} \). Thus, the inverse function \( g(y) = \frac{y - 3}{4} \) allows us to recover \( x \) from \( y \).
Verification is key to ensuring accuracy:

• By substituting \( g(y) = \frac{y - 3}{4} \) into \( f \), we confirm \( f(g(y)) = y \).
• Similarly, substituting \( f(x) \) into \( g \), we find \( g(f(x)) = x \).

This mutual recovery confirms the correctness of our inverse function, hence proving \( f \) is bijective and invertible.