Problem 112
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 \(f: R \rightarrow R\) be any function. Define \(g: R \rightarrow R\) by \(g(x)=|f(x)|\) for all \(x\). Then, \(g\) is continuous if \(f\) is continuous. Reason: Composition of two continuous functions is continuous
Step-by-Step Solution
Verified Answer
(B) Assertion is True, Reason is True; Reason is not a correct explanation for Assertion.
1Step 1: Analyze the Assertion
The assertion states that if function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is continuous, and we define \( g(x) = |f(x)| \), then \( g \) is also continuous. This is because the absolute value function \( |x| \) is continuous over \( \mathbb{R} \), and the composition of continuous functions (\( |f(x)| \)) remains continuous, specifically for absolute values which preserve continuity.
2Step 2: Analyze the Reason
The reason states that the composition of two continuous functions is continuous. This is generally true; when two functions \( f \) and \( g \) are continuous at a point or over their entire domains, \( f(g(x)) \) will also be continuous at that point or over the domains.
3Step 3: Evaluate the Connection between Assertion and Reason
While the reason asserts a general statement about the composition of continuous functions, it is not directly applicable to the assertion which involves a different principle. The assertion concerns the absolute value operation applied to \( f(x) \), rather than a simple composition. Thus, although both statements are true, the reason does not correctly explain the assertion.
Key Concepts
Composition of FunctionsAbsolute Value PropertiesMathematical Reasoning
Composition of Functions
When we talk about the composition of functions, we refer to the process of combining two functions to form a new function. In simpler terms, if you have two functions, say \( f(x) \) and \( g(x) \), composing them involves applying one function to the result of the other. This is usually represented as \( f(g(x)) \).
For the composition of functions to be continuous, both original functions \( f(x) \) and \( g(x) \) must be continuous themselves. This means that if no "breaks" or "jumps" occur in \( f(x) \) and \( g(x) \) individually, putting them together in a composition won't introduce any new discontinuities.
- **Example:** - Consider \( f(x) = x^2 \) and \( g(x) = \sin(x) \). Applying \( f \) to \( g \) yields \( f(g(x)) = (\sin(x))^2 \). Both \( x^2 \) and \( \sin(x) \) are continuous everywhere, so their composition \( (\sin(x))^2 \) is continuous as well.
For the composition of functions to be continuous, both original functions \( f(x) \) and \( g(x) \) must be continuous themselves. This means that if no "breaks" or "jumps" occur in \( f(x) \) and \( g(x) \) individually, putting them together in a composition won't introduce any new discontinuities.
- **Example:** - Consider \( f(x) = x^2 \) and \( g(x) = \sin(x) \). Applying \( f \) to \( g \) yields \( f(g(x)) = (\sin(x))^2 \). Both \( x^2 \) and \( \sin(x) \) are continuous everywhere, so their composition \( (\sin(x))^2 \) is continuous as well.
Absolute Value Properties
The absolute value of a number is a way of describing its distance from zero on the number line, without considering direction. It's an essential concept in mathematics since it always results in a non-negative number. For example, both \(|3|\) and \(|-3|\) equal 3, emphasizing distance rather than the direction from zero.
The key property of the absolute value that applies here is its continuity. The function \(|x|\) is continuous everywhere on the real number line \( \mathbb{R} \).
- **More on the Absolute Value Function:** - \(|x| = x\) when \(x \geq 0\) - \(|x| = -x\) when \(x < 0\)This characteristic ensures that regardless of the input from another continuous function \(f(x)\), the operation \(|f(x)|\) will result in a continuous function. Thus, any function transformed by the absolute value retains its continuity. It smooths out things by converting any dips below zero to positive, thus maintaining a "continuous" path.
The key property of the absolute value that applies here is its continuity. The function \(|x|\) is continuous everywhere on the real number line \( \mathbb{R} \).
- **More on the Absolute Value Function:** - \(|x| = x\) when \(x \geq 0\) - \(|x| = -x\) when \(x < 0\)This characteristic ensures that regardless of the input from another continuous function \(f(x)\), the operation \(|f(x)|\) will result in a continuous function. Thus, any function transformed by the absolute value retains its continuity. It smooths out things by converting any dips below zero to positive, thus maintaining a "continuous" path.
Mathematical Reasoning
Mathematical reasoning involves making logical connections between ideas and verifying their correctness. It ensures conclusions are based on sound reasoning and precise principles.
In the context of continuity and function composition, reasoning requires:- Understanding the specific characteristics of functions involved.- Applying the correct mathematical theorems or principles to establish the properties of these functions.
For example, when assessing the assertion that \(g(x) = |f(x)|\) is continuous if \(f(x)\) is continuous, it’s important to understand:- The continuity of \(f(x)\) itself.- The inherent continuity of the absolute value function.- How the operation of absolute value affects continuity.
The reason should validate this process by providing a logical connection. However, in our assertion, while the consistent nature of function composition is mentioned, it does not perfectly explain the assertion because the absolute value is not merely composition but involves a separate continuous transformation, distinct from general composition rules. Proper reasoning distinguishes between different scenarios to verify the truth with accuracy.
In the context of continuity and function composition, reasoning requires:- Understanding the specific characteristics of functions involved.- Applying the correct mathematical theorems or principles to establish the properties of these functions.
For example, when assessing the assertion that \(g(x) = |f(x)|\) is continuous if \(f(x)\) is continuous, it’s important to understand:- The continuity of \(f(x)\) itself.- The inherent continuity of the absolute value function.- How the operation of absolute value affects continuity.
The reason should validate this process by providing a logical connection. However, in our assertion, while the consistent nature of function composition is mentioned, it does not perfectly explain the assertion because the absolute value is not merely composition but involves a separate continuous transformation, distinct from general composition rules. Proper reasoning distinguishes between different scenarios to verify the truth with accuracy.
Other exercises in this chapter
Problem 110
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
View solution Problem 111
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
View solution Problem 112
If \(f(x)=\left\\{\begin{array}{cc}\frac{x\left(3 e^{1 / x}+4\right)}{2-e^{1 / x}}, x \neq 0 \\ 0 & , x=0\end{array}\right.\), then \(f(x)\) is (A) continuous a
View solution Problem 113
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
View solution