Problem 102 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Determine the following functions:- i. \(\quad f_{1}(x)=|x-1|\); ii. \(\quad f_{2}(x)=|2 x+1|\); iii. \(f_{3}(x)=|\ln x|\); iv. \(\quad f_{4}(x)=\left|e^{x}\right|\); v. \(f_{5}(x)=\operatorname{sgn}(x+3)\); vi. \(f_{6}(x)=\operatorname{sgn}(3 x+2)\); vii. \(f_{7}(x)=\operatorname{sgn}(\ln x)\); viii. \(f_{8}(x)=\operatorname{sgn}\left(e^{x}\right)\). \\{Ans. i. \(f_{1}(x)=x-1, \quad x \geq 1\) \(=1-x, \quad x<1\) ii. \(f_{2}(x)=2 x+1, \quad x \geq-\frac{1}{2}\) \(=-2 x-1, \quad x<-\frac{1}{2}\) iii. \(f_{3}(x)=\ln x, \quad x \geq 1\) \(=-\ln x, \quad 0-3\) \(=0, \quad x=-3\) \(=-1, \quad x<-3\) vi. \(f_{6}(x)=1, \quad x>-\frac{2}{3}\) \(=0, \quad x=-\frac{2}{3}\) \(=-1, \quad x<-\frac{2}{3}\) vii. \(f_{7}(x)=1, \quad x>1\) \(=0, \quad x=1\) \(=-1, \quad 0

Step-by-Step Solution

Answer

Ans. i. \(f_1(x)=x-1, \quad x \geq 1\) \(=1-x, \quad x<1\) ii. \(f_2(x)=2 x+1, \quad x \geq-\frac{1}{2}\) \(-2 x-1, \quad x<-\frac{1}{2}\) iii. \(f_3(x)=\ln x, \quad x \geq 1\) \(-\ln x, \quad 0-3\) \(=0, \quad x=-3\) \(-1, \quad x<-3\) vi. \(f_6(x)=1, \quad x>-\frac{2}{3}\) \(=0, \quad x=-\frac{2}{3}\) \(-1, \quad x<-\frac{2}{3}\) vii. \(f_7(x)=1, \quad x>1\) \(=0, \quad x=1\) \(-1, \quad 0

Step 1: Function \(f_1(x) = |x-1|\)

For the function \(f_1(x) = |x-1|\), we need to analyze two cases: 1. When \(x \geq 1\), this means that \(x-1\) is non-negative, so \(f_1(x) = x-1\). 2. When \(x < 1\), this means that \(x-1\) is negative, so we need to take the absolute value: \(f_1(x) = -(x-1) = 1-x\).

Step 2: Function \(f_2(x) = |2x+1|\)

For the function \(f_2(x) = |2x+1|\), we again have two cases to consider: 1. When \(x \geq -\frac{1}{2}\), that is, \(2x+1\) is non-negative and the absolute value leaves the expression unchanged: \(f_2(x) = 2x+1\). 2. When \(x < -\frac{1}{2}\), this means that \(2x+1\) is negative, so we need to take the absolute value: \(f_2(x) = -(2x+1) = -2x-1\).

Step 3: Function \(f_3(x) = |\ln x|\)

For the function \(f_3(x) = |\ln x|\), we need to consider two cases: 1. When \(x \geq 1\), this means that \(\ln x\) is non-negative, so \(f_3(x) = \ln x\). 2. When \(0 < x < 1\), this means that \(\ln x\) is negative, so we need to take the absolute value: \(f_3(x) = -\ln x\).

Step 4: Function \(f_4(x) = |e^x|\)

Since the function \(e^x\) is always positive for any value of \(x\), the absolute value will not change it. Therefore, the function remains as \(f_4(x) = e^x\).

Step 5: Function \(f_5(x) = \operatorname{sgn}(x+3)\)

The signum function has three possible values: 1. When \(x > -3\), it means that \(x+3\) is positive, and the signum function returns 1: \(f_5(x) = 1\). 2. When \(x = -3\), it means that \(x+3\) is zero, and the signum function returns 0: \(f_5(x) = 0\). 3. When \(x < -3\), it means that \(x+3\) is negative, and the signum function returns -1: \(f_5(x) = -1\).

Step 6: Function \(f_6(x) = \operatorname{sgn}(3x+2)\)

The signum function has three possible values: 1. When \(x > -\frac{2}{3}\), it means that \(3x+2\) is positive, and the signum function returns 1: \(f_6(x) = 1\). 2. When \(x = -\frac{2}{3}\), it means that \(3x+2\) is zero, and the signum function returns 0: \(f_6(x) = 0\). 3. When \(x < -\frac{2}{3}\), it means that \(3x+2\) is negative, and the signum function returns -1: \(f_6(x) = -1\).

Step 7: Function \(f_7(x) = \operatorname{sgn}(\ln x)\)

The signum function has three possible values: 1. When \(x > 1\), it means that \(\ln x\) is positive, and the signum function returns 1: \(f_7(x) = 1\). 2. When \(x = 1\), it means that \(\ln x\) is zero, and the signum function returns 0: \(f_7(x) = 0\). 3. When \(0 < x < 1\), it means that \(\ln x\) is negative, and the signum function returns -1: \(f_7(x) = -1\).

Step 8: Function \(f_8(x) = \operatorname{sgn}(e^x)\)

Because \(e^x\) is always positive for any value of \(x\), the signum function will always return 1: \(f_8(x) = 1\).

Key Concepts

Signum Function

The signum function, often denoted as \( \operatorname{sgn}(x) \), is a simple yet pivotal function in mathematics. Its primary purpose is to determine the sign of a real number. Though it sounds trivial, understanding its behavior is crucial in various mathematical contexts. The function operates based on three rules:

• If \( x > 0 \), then \( \operatorname{sgn}(x) = 1 \).
• If \( x = 0 \), then \( \operatorname{sgn}(x) = 0 \).
• If \( x < 0 \), then \( \operatorname{sgn}(x) = -1 \).

The signum function can be visualized as a stepwise function that sharply changes values at \( x = 0 \). Its applications range from simplifying conditional expressions to solving problems in control theory and signal processing. By reducing a complex piecewise function to just three potential outputs, \( \operatorname{sgn}(x) \) makes the analysis of sign-based systems straightforward.

Piecewise Functions

Piecewise functions are fascinating because they can represent a diverse set of behaviors within a single function. Instead of using one rule to define the function, multiple expressions define its behavior over different intervals. This flexibility is particularly useful when different pieces of a function need to address specific situations or conditions.
Here are a few important points about piecewise functions:

• Each "piece" has its own formula and is applicable over a specific interval.
• The domain of the function is divided into these intervals, often separated by "breakpoints," where the behavior of the function changes.
• They can be continuous, where the pieces connect smoothly, or discontinuous, where there might be sudden jumps.

For example, consider \( f(x) = \begin{cases} x-1 & \text{if } x \geq 1 \ 1-x & \text{if } x < 1 \end{cases} \). This function changes its rule depending on whether \( x \) is greater than or less than \( 1 \).
Piecewise functions are crucial in modeling real-world situations where different rules or behaviors apply over different situations, such as tax brackets, shipping costs, and various pricing models.

Logarithmic Functions

Logarithmic functions are inverse exponential functions and fundamentally express the idea of a logarithm as the power to which a number must be raised to obtain another number. The most common logarithms used are base 10, known as common logs, and base \( e \), known as natural logs (denoted as \( \ln \)). Their significance arises in fields ranging from natural sciences to information theory.
Core aspects of logarithmic functions include:

• The domain of \( \ln(x) \) is \( (0, \infty) \), meaning it is defined only for positive real numbers.
• The function is continuous and increases gradually, passing through the point \( (1,0) \), since \( \ln(1) = 0 \).
• As \( x \to 0^{+} \), the function tends to \( -\infty \), and as \( x \to \infty \), \( \ln(x) \to \infty \).
• It shows the "compounded growth" rate of processes where the time taken for a complete fact to become visible is proportionate to the amount already known.

When working with logarithmic expressions and equations, always consider the domain restrictions and be cautious of applying rules. Common applications include systems involving multiplicative processes like sound intensity (decibels), pH in chemistry, and more.

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Problem 101

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Problem 103

Other exercises in this chapter

Problem 100

$$ \text { Determine the function } f(x)=|x| \times \operatorname{sgn} x .\\{\text { Ans. } f(x)=x\\} $$

View solution

Problem 101

Determine the following functions:- i. \(\quad f_{1}(x)=\sin |x|\); ii. \(\quad f_{2}(x)=\sin ^{-1}(\operatorname{sgn} x)\); iii. \(f_{3}(x)=|x|^{2}\); iv. \(\q

View solution

Problem 103

\begin{aligned} &\text { Given } f(x)=x+2, \quad-2 \leq x \leq 0\\\ &=x-1, \quad 0

View solution

Problem 104

$$ \begin{aligned} &\begin{aligned} f(x) &=x^{2}, \quad x

View solution

Key Concepts

Signum Function

The signum function, often denoted as \( \operatorname{sgn}(x) \), is a simple yet pivotal function in mathematics. Its primary purpose is to determine the sign of a real number. Though it sounds trivial, understanding its behavior is crucial in various mathematical contexts. The function operates based on three rules:

• If \( x > 0 \), then \( \operatorname{sgn}(x) = 1 \).
• If \( x = 0 \), then \( \operatorname{sgn}(x) = 0 \).
• If \( x < 0 \), then \( \operatorname{sgn}(x) = -1 \).

The signum function can be visualized as a stepwise function that sharply changes values at \( x = 0 \). Its applications range from simplifying conditional expressions to solving problems in control theory and signal processing. By reducing a complex piecewise function to just three potential outputs, \( \operatorname{sgn}(x) \) makes the analysis of sign-based systems straightforward.

Piecewise Functions

Piecewise functions are fascinating because they can represent a diverse set of behaviors within a single function. Instead of using one rule to define the function, multiple expressions define its behavior over different intervals. This flexibility is particularly useful when different pieces of a function need to address specific situations or conditions.
Here are a few important points about piecewise functions:

• Each "piece" has its own formula and is applicable over a specific interval.
• The domain of the function is divided into these intervals, often separated by "breakpoints," where the behavior of the function changes.
• They can be continuous, where the pieces connect smoothly, or discontinuous, where there might be sudden jumps.

For example, consider \( f(x) = \begin{cases} x-1 & \text{if } x \geq 1 \ 1-x & \text{if } x < 1 \end{cases} \). This function changes its rule depending on whether \( x \) is greater than or less than \( 1 \).
Piecewise functions are crucial in modeling real-world situations where different rules or behaviors apply over different situations, such as tax brackets, shipping costs, and various pricing models.

Logarithmic Functions

Logarithmic functions are inverse exponential functions and fundamentally express the idea of a logarithm as the power to which a number must be raised to obtain another number. The most common logarithms used are base 10, known as common logs, and base \( e \), known as natural logs (denoted as \( \ln \)). Their significance arises in fields ranging from natural sciences to information theory.
Core aspects of logarithmic functions include:

• The domain of \( \ln(x) \) is \( (0, \infty) \), meaning it is defined only for positive real numbers.
• The function is continuous and increases gradually, passing through the point \( (1,0) \), since \( \ln(1) = 0 \).
• As \( x \to 0^{+} \), the function tends to \( -\infty \), and as \( x \to \infty \), \( \ln(x) \to \infty \).
• It shows the "compounded growth" rate of processes where the time taken for a complete fact to become visible is proportionate to the amount already known.

When working with logarithmic expressions and equations, always consider the domain restrictions and be cautious of applying rules. Common applications include systems involving multiplicative processes like sound intensity (decibels), pH in chemistry, and more.