Problem 10 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f: R \rightarrow R\) be any function. Define \(g: R \rightarrow R\) by \(g(x)\) \(=|f(x)|\) for all \(x\). Then \(g\) is (A) onto if \(f\) is onto (B) one-one if \(f\) is one-one (C) continuous if \(f\) is continuous (D) differentiable if \(f\) is differentiable

Step-by-Step Solution

Answer

Option (C) is true: \( g \) is continuous if \( f \) is continuous.

Step 1: Examine Option (A)

To determine if \( g \) is onto if \( f \) is onto, recall that a function \( g \) is onto if every element in its codomain is the image of some element in its domain. Given \( g(x) = |f(x)| \), the range of \( g \) is \([0, +\infty)\) because absolute values are always non-negative. Thus, even if \( f \) is onto, \( g \) will always map to non-negative values, never achieving negative values. Hence, (A) is false.

Step 2: Examine Option (B)

A function \( g \) is one-one (injective) if different inputs map to different outputs. If \( f \) is one-one, there exist distinct inputs such that \( f(x_1) eq f(x_2) \). However, if \( |f(x_1)| = |f(x_2)| \), \( g \) will not be one-one because different inputs can map to the same output after taking the absolute value. Thus, (B) is false.

Step 3: Examine Option (C)

Check whether \( g \) is continuous if \( f \) is continuous. A function with an absolute value form, such as \( g(x) = |f(x)| \), is continuous whenever the function inside, \( f(x) \), is continuous as absolute value operations preserve continuity. Hence, (C) is true.

Step 4: Examine Option (D)

Determine if \( g \) is differentiable if \( f \) is differentiable. The absolute value function, \( |f(x)| \), may introduce points of non-differentiability, particularly if \( f(x) = 0 \), as the slope from the left might not equal the slope from the right. Therefore, if \( f \) is differentiable, \( g \) might not be, particularly at points where \( f(x) = 0 \). Thus, (D) is false.

Key Concepts

Continuity

In mathematics, continuity is a fundamental property of functions. When we say a function is continuous, we imply that its graph can be drawn without lifting the pencil from the paper. Mathematically, a function \( f(x) \) is continuous at a point \( x = c \) if:

• \( f(c) \) is defined.
• The limit of \( f(x) \) as \( x \) approaches \( c \), denoted as \( \lim_{{x \to c}} f(x) \), exists.
• \( \lim_{{x \to c}} f(x) = f(c) \).

For an absolute value function, like \( g(x) = |f(x)| \), continuity at any point depends on the continuity of \( f(x) \) itself.
If \( f(x) \) is continuous, then \( g(x) \), which involves an absolute value operation, maintains this continuity. This is because absolute value operations do not disrupt the continuity of \( f(x) \).
So, if you confirm the continuity of \( f(x) \), you can confidently say that \( g(x) = |f(x)| \) is continuous as well.

Differential Calculus

Differential calculus focuses on understanding how functions change. It involves concepts such as derivatives, which represent the rate of change or the slope of a function.
For a function \( f(x) \) to be differentiable at a point \( x = c \), the derivative \( f'(c) \) must exist. This requires the function to be smooth, without sharp corners or points where the slope suddenly changes.When considering a function \( g(x) = |f(x)| \), differentiability can become tricky. The absolute value function can introduce points where the derivative does not exist, especially where \( f(x) = 0 \).
At these points, \( g(x) \) could have a sharp corner, resulting in different slopes approaching from the left and right. Therefore, even if \( f(x) \) is differentiable, \( g(x) \) might not be differentiable at certain points, specifically at those where \( f(x) = 0 \).
This highlights the significance of examining the behavior of \( f(x) \) around zero before concluding the differentiability of \( g(x) \).

Injective Functions

Injective functions, also known as one-to-one functions, map distinct inputs to distinct outputs.
This means for a function \( f(x) \), if \( x_1 eq x_2 \), then \( f(x_1) eq f(x_2) \). This uniqueness ensures no two different inputs have the same output.However, when dealing with \( g(x) = |f(x)| \), the absolute value can impede this injectivity. If \( f(x_1) = -f(x_2) \), the application of the absolute value results in \( |f(x_1)| = |f(x_2)| \).
This means different inputs can correspond to the same output in \( g(x) \). Therefore, \( g(x) = |f(x)| \) fails to be injective, even if \( f(x) \) was initially one-to-one. This property is crucial to consider when analyzing the characteristics of transformed functions like \( g(x) \).
To ensure injectivity after transformation, any operation altering the uniqueness of outputs, such as the absolute value, must be critically evaluated.