Problem 1 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Define the following analytical functions using basic functions:- i. \(\quad f(x)=\sin x+\ln x, x>1\) ii. \(\quad f(x)=x^{2}-\cos x, x \geq 0\) iii. \(\quad|x|\) iv. \(\operatorname{sgn} x\) v. \(\quad f(x)=e^{x}-\sin x, x<0\) \(=3 \sqrt{x}, \quad x \geq 0\) vi. \(\quad f(x)=\frac{\sin x}{\sqrt{\ln x}+\cosh x}\) vii. \(f(x)=\frac{\sqrt{e^{x}}}{1+x^{2}}, x \leq 0\) \(=\frac{\frac{1}{x}+\ln x}{e^{x}+\ln ^{2} x}, x>0\) viii. \(f(x)=\sqrt{\ln (\sin x)+\sin \sqrt{\ln x}}\) ix. \(\quad f(x)=\left(\frac{x}{1+\sin x}\right)^{3}\) x. \(\quad f(x)=2^{\cos x+\sqrt{x}}\) xi. \(f(x)=x^{x}\) xii. \(f(x)=(\sin x)^{\cos x}\) xiii. \(f(x)=\left(\frac{1+x}{1-x}\right)^{x}\) xiv. \(f(x)=\sin ^{-1}\left(\ln x+\sin ^{-1} x\right)\) xv. \(\quad f(x)=\cos ^{-1}\left(x \ln x+\sqrt{\tan ^{-1} \sqrt{x}}\right)\) xvi. \(f(x)=\log _{\sin x} \cos x\)

Step-by-Step Solution

Answer

In this problem, we have to define a list of analytical functions in terms of basic functions. Here is a list of the functions defined: i. \(f(x)=\sin x+\ln x, x>1\) ii. \(f(x)=x^{2}-\cos x, x \geq 0\) iii. \(|x|\) iv. \(\operatorname{sgn} x\) v. \(f(x)=e^{x}-\sin x, x<0\) vi. \(f(x)=\frac{\sin x}{\sqrt{\ln x}+\cosh x}\) vii. \(f(x)=\frac{\sqrt{e^{x}}}{1+x^{2}}, x \leq 0\) viii. \(f(x)=\sqrt{\ln (\sin x)+\sin \sqrt{\ln x}}\) ix. \(f(x)=\left(\frac{x}{1+\sin x}\right)^{3}\) x. \(f(x)=2^{\cos x+\sqrt{x}}\) xi. \(f(x)=x^{x}\) xii. \(f(x)=(\sin x)^{\cos x}\) xiii. \(f(x)=\left(\frac{1+x}{1-x}\right)^{x}\) xiv. \(f(x)=\sin ^{-1}\left(\ln x+\sin ^{-1} x\right)\) xv. \(f(x)=\cos ^{-1}\left(x \ln x+\sqrt{\tan ^{-1} \sqrt{x}}\right)\) xvi. \(f(x)=\log _{\sin x} \cos x\)

Step 1: i. Defining \(f(x)=\sin x+\ln x, x>1\)

This function is the sum of the sine function and the natural logarithm function. The function is defined for values of x greater than 1.

Step 2: ii. Defining \(f(x)=x^{2}-\cos x, x \geq 0\)

This function is the difference between a square function and the cosine function. The function is defined for values of x greater than or equal to 0.

Step 3: iii. Defining \( |x|\)

This function is the absolute value function. It is equal to the value of x if x is positive or zero, and is equal to the negative of x if x is negative.

Step 4: iv. Defining \(\operatorname{sgn} x\)

The sign function gives the sign of a real number x, returning -1 for negative numbers, 1 for positive numbers and 0 for zero.

Step 5: v. Defining \(f(x)=e^{x}-\sin x, x<0\)

This function is the difference between an exponential function and the sine function. The function is defined for x less than 0.

Step 6: vi. Defining \(f(x)=\frac{\sin x}{\sqrt{\ln x}+\cosh x}\)

This function involves a ratio of the sine function to the sum of a square root function of the natural logarithm and a hyperbolic cosine function.

Step 7: vii. Defining \(f(x)=\frac{\sqrt{e^{x}}}{1+x^{2}}, x \leq 0\)

This function involves a ratio of a square root function of the exponential function to the sum of 1 and the square function. The function is defined for x less than or equal to 0.

Step 8: viii. Defining \(f(x)=\sqrt{\ln (\sin x)+\sin \sqrt{\ln x}}\)

This function contains square root, natural logarithmic, sine functions and also functions which are composite functions.

Step 9: ix. Defining \(f(x)=\left(\frac{x}{1+\sin x}\right)^{3}\)

This function is the cube of the ratio of x to the sum of 1 and the sine function.

Step 10: x. Defining \(f(x)=2^{\cos x+\sqrt{x}}\)

This function is a power function where the power is the sum of the cosine function and a square root function.

Step 11: xi. Defining \(f(x)=x^{x}\)

This function is the power function where the base and the exponent are both x.

Step 12: xii. Defining \(f(x)=(\sin x)^{\cos x}\)

This function is a power function where the sine function is the base and the cosine function is the exponent.

Step 13: xiii. Defining \(f(x)=\left(\frac{1+x}{1-x}\right)^{x}\)

This function is a power function where the exponent is x and the base is the ratio of the sum of 1 and x to the difference of 1 and x.

Step 14: xiv. Defining \(f(x)=\sin ^{-1}\left(\ln x+\sin ^{-1} x\right)\)

This function is the inverse sine function of the sum of the natural logarithm function and the inverse sine function of x.

Step 15: xv. Defining\(f(x)=\cos ^{-1}\left(x \ln x+\sqrt{\tan ^{-1} \sqrt{x}}\right)\)

This is the inverse cosine function applied to a complex composite function which involves natural logarithm, multiplication, square root, and inverse tangent functions.

Step 16: xvi. Defining \(f(x)=\log _{\sin x} \cos x\)

This function is the logarithm function, where \(\sin x\) is the base of the logarithm and \(\cos x\) is used as an argument.

Key Concepts

basic functions

Basic functions are the building blocks of more complex equations and operations in mathematics. They are defined as functions that are very simple in nature, such as linear functions, quadratic functions, constants, and other polynomial functions.
Understanding basic functions is crucial because it helps set the foundation for more advanced math concepts.

• Linear functions: These are functions where the graph forms a straight line, expressed in the form \( f(x) = mx + c \).
• Quadratic functions: Functions represented as \( f(x) = ax^2 + bx + c \), showing a parabolic curve on the graph.
• Constants: These are functions that return the same value no matter the input, written as \( f(x) = c \).

These basic functions serve as the basis for creating more complex ones by combining them using operations like addition, subtraction, multiplication, division, and composition of functions.

trigonometric functions

Trigonometric functions are functions that relate the angles of a triangle to the lengths of its sides. They are critical in studying periodic phenomena, waves, and oscillations. The most common trigonometric functions include sine, cosine, and tangent, often abbreviated as sin, cos, and tan, respectively.

• Sine Function: Defined as \( \sin(\theta) = \frac{opposite}{hypotenuse} \); calculates the y-coordinate of a point on the unit circle.
• Cosine Function: Defined as \( \cos(\theta) = \frac{adjacent}{hypotenuse} \); calculates the x-coordinate of a point on the unit circle.
• Tangent Function: Defined as \( \tan(\theta) = \frac{opposite}{adjacent} \); represents the slope of the line produced by the angle.

These functions are pivotal in defining waveforms and are extensively used in physics, engineering, and many other scientific domains.

logarithmic functions

Logarithmic functions are the inverse operations of exponential functions. They play a significant role in solving equations where the variable is an exponent. A logarithmic function is generally expressed in the form \( y = \log_b(x) \), where \( b \) is the base of the logarithm.

• Natural Logarithm: Written as \( \ln(x) \), it uses the mathematical constant \( e \) (approximately 2.718) as its base. It is vital in calculus and appears frequently in mathematical modeling.
• Common Logarithm: Written as \( \log_{10}(x) \), implies the base 10, and is commonly used in measuring sound intensity (decibels) and in scientific calculations.
• Change of Base Formula: Converts logarithms from one base to another, given by \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \), where \( k \) is a new base.

Logarithmic functions are crucial in fields like finance and sciences where growth rates and exponential changes need to be analyzed.

inverse trigonometric functions

Inverse trigonometric functions help in finding angles where the trigonometric function values are known. These functions are particularly useful in applications requiring angle calculations where distances are known.

• Inverse Sine Function: Expressed as \( \sin^{-1}(x) \) or arcsin, it provides the angle whose sine is \( x \).
• Inverse Cosine Function: Denoted as \( \cos^{-1}(x) \) or arccos, it returns the angle whose cosine is \( x \).
• Inverse Tangent Function: Notated \( \tan^{-1}(x) \) or arctan, it finds the angle whose tangent is \( x \).

These functions are essential in setting up angles needed for navigating practical applications, such as engineering tasks, physics calculations, and even computer visualizations.