Problem 53 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Plot graph of the following functions:- i. \(y=e^{\ln |x|}\). ii. \(y=x^{\log _{x} 2}\). iii. \(y=\operatorname{sgn}(\sin x)\). iv. \(y=x^{\operatorname{sgn} x}\). v. \(\quad y=\operatorname{sgn}\left(\tan ^{-1} x\right)\). vi. \(y=(\operatorname{sgn} x)^{\operatorname{sgn} x}\). vii. \(y=\sin ^{-1}(\operatorname{sgn} x)\). viii. \(y=\cos ^{-1}(\operatorname{sgn} x)\). ix. \(\quad y=|x-1|+|x-2|+x\). x. \(\quad y=2^{\frac{|x|+x}{x}}\). xi. \(y=\log _{x} x^{2}\). xii. \(y=\ln \tan x+\ln \cot x\).

Step-by-Step Solution

Answer

In this exercise, we analyzed and plotted the following functions: i. \[y=e^{\ln |x|}\] has a domain of all real numbers except x=0 and is symmetric about the y-axis. ii. \[y=x^{\log _{x} 2}\] has a domain of all positive real numbers and is an increasing function. iii. \[y=\operatorname{sgn}(\sin x)\] has a domain of all real numbers and follows square-wave behavior. iv. \[y=x^{\operatorname{sgn} x}\] has a domain of all real numbers except x=0 and shows an increasing function on the right side of the origin and a constant function on the left side. v. \[y=\operatorname{sgn}\left(\tan ^{-1} x\right)\] has a domain of all real numbers and shows square-wave behavior between 1 and -1. vi. \[y=(\operatorname{sgn} x)^{\operatorname{sgn} x}\] has a domain of all real numbers except x=0 and shows two horizontal lines at y=1 for x>0 and y=-1 for x<0. vii. \[y=\sin ^{-1}(\operatorname{sgn} x)\] has a domain of all real numbers except x=0 and shows two horizontal lines at y=π/2 for x>0 and y=-π/2 for x<0. viii. \[y=\cos ^{-1}(\operatorname{sgn} x)\] has a domain of all real numbers except x=0 and shows two horizontal lines at y=0 for x>0 and y=π for x<0. ix. \[y=|x-1|+|x-2|+x\] has a domain of all real numbers and shows a V-shaped pattern. x. \[y=2^{\frac{|x|+x}{x}}\] has a domain of all real numbers except x=0 and shows a hole at x=0, with the function value equal to 1 for x<0 and 4 for x>0. xi. \[y=\log _{x} x^{2}\] has a domain of positive real numbers except x=1 and shows upward/downward concave curve. xii. \[y=\ln \tan x+\ln \cot x\] has a domain of real numbers between -π/2 and π/2, and it shows a wave-like shape between vertical asymptotes at x=-π/2, 0, and π/2.

Step 1: Function properties

For the function \(y=e^{\ln |x|}\), its domain is all real numbers except for x=0 since natural logarithm is undefined at 0. This function is positive for all x and is symmetric about the y-axis.

Step 2: Sketch the graph

To sketch the graph, observe that it's an exponential function. It will appear as the graph of \(y=e^x\) but with additional symmetry. It will have a vertical asymptote at x=0, where it approaches infinity on either side of the asymptote. #ii. Function ii.#

Step 3: Function properties

For the function \(y=x^{\log _{x} 2}\), the domain is all real positive numbers since logarithm is only defined for positive numbers. The base of the exponent is x, and the exponent is the logarithm in base x of 2.

Step 4: Sketch the graph

The function will have a range of all positive numbers. It's an increasing function with a slow growth with x. So, the graph will slowly increase as x increases. #iii. Function iii.#

Step 5: Function properties

For the function \(y=\operatorname{sgn}(\sin x)\), the domain is all real numbers. The function equals to 1 when the sine function is positive and -1 when the sine function is negative.

Step 6: Sketch the graph

The graph will follow the behavior of the sine function, but remain on either +1 or -1. It will have square-wave behavior. #iv. Function iv.#

Step 7: Function properties

For the function \(y=x^{\operatorname{sgn} x}\), the domain is all real numbers. The function is equal to \(x^1=x\) for x>0 and \(x^0=1\) for x<0. At x=0, the function is undefined.

Step 8: Sketch the graph

The graph will show an increasing function on the right side of the origin and a constant horizontal line at y=1 on the left side of the origin, with a hole at x=0. #v. Function v.#

Step 9: Function properties

For the function \(y=\operatorname{sgn}\left(\tan ^{-1} x\right)\), the domain is all real numbers. The function value is equal to the sign of the output of the inverse tangent function.

Step 10: Sketch the graph

The graph will have a square-wave behavior between 1 and -1. The value will alternate between 1 and -1 based on the sign of the arctan function. #vi. Function vi.#

Step 11: Function properties

For the function \(y=(\operatorname{sgn} x)^{\operatorname{sgn} x}\), the domain is all real numbers except x=0. The function value is either 1 or -1.

Step 12: Sketch the graph

The graph will be a horizontal line with a hole at x=0. It will appear as two horizontal lines, one at y=1 for x>0 and another at y=-1 for x<0. #vii. Function vii.#

Step 13: Function properties

For the function \(y=\sin ^{-1}(\operatorname{sgn} x)\), the domain is all real numbers except x=0. The range is between -π/2 and π/2.

Step 14: Sketch the graph

The graph will be two horizontal lines with a hole at x=0, y=π/2 for x>0 and y=-π/2 for x<0. #viii. Function viii.#

Step 15: Function properties

For the function \(y=\cos ^{-1}(\operatorname{sgn} x)\), the domain is all real numbers except x=0. The range is between 0 and π.

Step 16: Sketch the graph

The graph will be two horizontal lines with a hole at x=0, y=0 for x>0 and y=π for x<0. #ix. Function ix.#

Step 17: Function properties

For the function \(y=|x-1|+|x-2|+x\), the domain is all real numbers.

Step 18: Sketch the graph

The graph will show a V-shaped pattern due to the combination of the absolute values. As x increases, the graph will have an upwards slope. #x. Function x.#

Step 19: Function properties

For the function \(y=2^{\frac{|x|+x}{x}}\), the domain is all real numbers except for x=0 since division by zero is undefined.

Step 20: Sketch the graph

The point x=0 will be a hole in the graph. For x<0, the exponent equals 0, and the function value is 1. For x>0, the exponent equals 2, and the function value is 4. #xi. Function xi.#

Step 21: Function properties

For the function \(y=\log _{x} x^{2}\), the domain is positive real numbers except x=1 since the logarithm is undefined at x=1.

Step 22: Sketch the graph

The graph will have a hole at x=1. For x<1, it will have an upward concave curve, while for x>1, it will have a downward concave curve. The range is all positive real numbers. #xii. Function xii.#

Step 23: Function properties

For the function \(y=\ln \tan x+\ln \cot x\), the domain is real numbers between -π/2 and π/2, excluding 0.

Step 24: Sketch the graph

The graph will have vertical asymptotes at x=-π/2, 0, and π/2. The function value is 0 when x=-π/4 and π/4. The graph will be a wave-like shape between the asymptotes.

Key Concepts

Understanding the Sign Function

The sign function, denoted as \(\operatorname{sgn}(x)\), is a simple piece of mathematics used to determine the sign of a given real number. This function provides us with three possible outputs: -1, 0, or 1. If the input \(x\) is positive, the function returns 1. If \(x\) is negative, the function returns -1. Lastly, if \(x\) equals 0, the function result is 0. The sign function is crucial when dealing with functions that have behaviors dependent on the sign of \(x\), like in some of the piecewise functions you will encounter.
For example, the function \(y=\operatorname{sgn}(\sin x)\) represents a square wave function, dictating the sign based on the sine function oscillating between positive and negative values.

Domain and Range Basics

The concepts of domain and range are essential in understanding any type of function. The domain of a function refers to all the possible input values (or 'x' values) for which the function is defined. For instance, in the function \(y=e^{\ln |x|}\), the domain includes all real numbers except \(x=0\) because the natural logarithm is undefined at zero.
On the flip side, the range of a function reflects all possible output values (or 'y' values) the function can produce. Domains and ranges will vary significantly across different functions, and knowing how to determine these can significantly aid in graphing and analyzing the behavior of functions.

Exploring Logarithmic and Exponential Functions

Logarithmic and exponential functions are foundational in math, especially in understanding growth and decay processes. Exponential functions, like \(y=e^x\), describe situations that grow or decay at an exponential rate—they increase rapidly for large values of 'x.' Meanwhile, logarithmic functions are the inverse of exponential functions.
In the function \(y=x^{\log _{x} 2}\), you have a combination of both: a base 'x' raised to a logarithm with a base of 'x'. The domain remains positive real numbers since logarithms require positive inputs. These functions model many real-world phenomena, ranging from population dynamics to interest compounding.

Understanding Piecewise Functions

Piecewise functions are unique in that they are defined by different expressions across their domain. These functions can appear complicated but are simply a combination of several functions, each holding within a specific interval. Consider the function \(y=x^{\operatorname{sgn} x}\). Here, the behavior changes depending on whether the input \(x\) is positive or negative. Such functions often have discontinuities or defined breaks, showcasing different behaviors at those critical points.
Another example is \(y=|x-1|+|x-2|+x\), which could be viewed as a piecewise function due to the absolute value components. Plotting these requires identifying intervals and the respective expression to understand how a piecewise function behaves across the entire domain.