Problem 150 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: Let \(f\) and \(g\) be increasing and decreasing functions respectively from \([0, \infty]\) to \([0, \infty] .\) Let \(h(x)=f(g(x))\). If \(h(0)=0\), then \(h(x)\) is always zero Reason: \(h(x)\) is an increasing function of \(x\)

Step-by-Step Solution

Answer

Option (D): Assertion is False, Reason is False.

Step 1: Analyze the Assertion

First, we need to understand the assertion: It states that if \( h(x) = f(g(x)) \), where \( f \) is an increasing function and \( g \) is a decreasing function, and given \( h(0) = 0 \), then \( h(x) \) is always zero. This implies every \( h(x) \) for \( x \geq 0 \) should be zero if the assertion is true.

Step 2: Assess the behavior of functions \( f \), \( g \), and \( h \)

Since \( f \) is increasing, it means that as the input to \( f \) increases, the output cannot decrease. Since \( g \) is decreasing, as \( x \) increases, \( g(x) \) decreases. Thus, the composition \( h(x) = f(g(x)) \) is a function where the inner function decreases, affecting the input to the increasing function \( f \).

Step 3: Investigate \( h(x) \) given \( h(0) = 0 \)

Given \( h(0) = f(g(0)) = 0 \), this does not automatically imply \( f(g(x)) = 0 \) for all \( x \geq 0 \). Since \( g \) is decreasing, \( g(x) \) for \( x > 0 \) could still result in values for which \( f(g(x)) \) is not zero unless the specific forms of \( g \) and \( f \) always result in zero, which is not guaranteed here.

Step 4: Determine if the Assertion is True or False

Given the possibility that \( g(x) \) can vary for \( x > 0 \), and \( f(g(x)) \) can provide non-zero values under different circumstances, the assertion that \( h(x) \) is always zero is incorrect.

Step 5: Evaluate the Reason

The reason states that \( h(x) \) is an increasing function of \( x \). Since \( g \) is a decreasing function and modifying the input to \( f \), which is increasing, the composition \( h(x) = f(g(x)) \) is not necessarily increasing in terms of \( x \). Therefore, the reason is also false.

Step 6: Conclusion Based on Given Options

The assertion: \( h(x) \) is always zero is false, and the reason that \( h(x) \) is an increasing function is also false. Thus, the correct choice is option (D): Assertion(A) is False, Reason(R) is True.

Key Concepts

Function Composition

Function composition involves combining two functions so that the output of one function becomes the input of the other. If we have two functions, say \( f \) and \( g \), we can form a new function \( h(x) = f(g(x)) \). This means we first apply function \( g \) to \( x \), and then apply function \( f \) to the result of \( g(x) \).

This is a powerful concept because it allows us to build complex functions from simpler ones. Understanding how the individual functions behave is crucial in predicting the behavior of their composition. For instance:

• If \( f \) is increasing, then for any higher input, the output will not decrease.
• If \( g \) is decreasing, then as \( x \) increases, \( g(x) \) decreases.

When composing these, if \( g(x) \) provides decreasing values to the increasing function \( f \), the overall behavior of \( h(x) \) will depend on how these changes interact.

The concept of function composition is central in many fields such as calculus and real analysis, and it's often used to transform problems into a more workable form by changing their domain or simplifying their structure.

Increasing and Decreasing Functions

Functions can be classified by how their outputs respond to changes in the input.

• An increasing function is one where, if we increase the input, the output either stays the same or increases. In mathematical terms, for any two numbers \( a \) and \( b \) where \( a < b \), it will always be the case that \( f(a) \leq f(b) \).
• A decreasing function, on the other hand, will have outputs that decrease or stay the same as the input increases. Therefore, for any \( a < b \), we have \( g(a) \geq g(b) \).

This classification helps in understanding the overall behavior of composed functions, particularly in determining the range and value tendencies.

The influence of these characteristics was evident in our exercise, where \( f \) and \( g \) had opposing behaviors (increasing and decreasing), leading to a more complex function behavior in \( h(x) = f(g(x)) \). Specifically, it showed that even if each function can be understood individually, their interaction can drastically affect the output in non-intuitive ways when combined.

Assertion and Reason Problems

Assertion and reason problems are common in mathematics, where you're asked to determine the validity of a statement (the "assertion") and whether a given "reason" justifies it.

In these problems, you are often given a statement and a potential explanation, and you need to decide:

• Whether both the assertion and reason are true.
• If they are true, does the reason correctly explain the assertion?
• Whether only one of them is true or both are false.

This type of problem is excellent for developing critical thinking skills because it requires a deep understanding of the concepts involved, rather than just a surface-level reading.

In our exercise, the challenge was to understand not just if \( h(x) \) was always zero, but also if the reason provided about function behavior explained that assertion. Both parts were false in this case. Understanding why can reveal much about function behavior and the subtle complexities that come with function composition and individual function characteristics.