Problem 144 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The range of the function Column-I I. \(y=\log _{\sqrt{5}}\) \(\\{\sqrt{2}(\sin x-\cos x)+3\\}\) II. \(y=\log _{2}\left\\{2-\log _{\sqrt{3}}\right.\) \(\left.\left(16 \sin ^{2} x+1\right)\right\\}\) III. \(y=\frac{e^{x}-e^{-|x|}}{e^{x}+e^{|x|}}\) IV. \(y=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}, x \geq 0\) \begin{tabular}{l} Column-II \\ \hline (A) \([0,1)\) \end{tabular} (B) \((-\infty, 1]\) (C) \([0,2]\) (D) \(\left[0, \frac{1}{2}\right)\)

Step-by-Step Solution

Answer

I \((B)\), II \((A)\), III \((B)\), IV \((A)\).

Step 1: Analyze Function I

The function given is \(y = \log_{\sqrt{5}}\left(\sqrt{2}(\sin x - \cos x) + 3\right)\). First, analyze the expression inside the logarithm. The term \(\sqrt{2}(\sin x - \cos x)\) varies between \(-\sqrt{2}\) and \(\sqrt{2}\) due to the range of \(\sin x - \cos x\). Thus, the expression inside the log ranges from \(3 - \sqrt{2}\) to \(3 + \sqrt{2}\). The range of \(\log_{\sqrt{5}} z\) for \(z\in (3 - \sqrt{2}, 3 + \sqrt{2})\) would be \(y < 1\), which corresponds to \((-\infty, 1)\).

Step 2: Analyze Function II

The function is \(y = \log_{2}\left\{2 - \log_{\sqrt{3}} \left(16\sin^2 x + 1\right)\right\}\). First, consider \(16\sin^2 x\), whose range is \([0, 16]\). This makes \(16\sin^2 x + 1\) range from 1 to 17. The range of \(\log_{\sqrt{3}}(16\sin^2 x + 1)\) is \(log_{\sqrt{3}}[1, 17]\). Hence, \(z = 2 - \log_{\sqrt{3}}(16\sin^2 x + 1)\) varies between 2 and some minimum value greater than \(-\infty\), thus the range of \(\log_2 z\) aligns with \([0, 1)\).

Step 3: Analyze Function III

The function given is \(y = \frac{e^x - e^{-|x|}}{e^x + e^{|x|}}\). For both \(x \geq 0\) and \(x < 0\), this simplifies to the same expression as in Function IV, showing that \(y\) can vary between \(-1\) and 1. However, since it constantly aligns \(x \geq 0\), the range is confirmed as \((-\infty, 1]\).

Step 4: Analyze Function IV

The expression is \(y = \frac{e^x - e^{-x}}{e^x + e^{-x}}\) with \(x \geq 0\). This expression simplifies to \(\tanh(x)\), where the range of \(\tanh(x)\) is \([0, 1)\) for \(x \geq 0\).

Step 5: Match Columns

Now match the functions in Column I to the ranges given in Column II:- I corresponds to \((-\infty, 1]\).- II corresponds to \([0, 1)\).- III corresponds to \((-\infty, 1]\).- IV corresponds to \([0, 1)\).

Key Concepts

Logarithmic Functions

Logarithmic functions are fundamental in mathematics, especially when analyzing exponential growth or decay. The logarithm of a number is the power to which a base, like 10 or 2, must be raised to produce that number. For example, in the expression \(y = \log_b(x)\), \(b\) is the base, and \(x\) is the argument.
Here are some crucial points about logarithmic functions:

• **Domain**: The argument \(x\) must be greater than zero. Consequently, \(x > 0\) for the logarithm to be defined.
• **Range**: A logarithmic function's range is all real numbers. It can approach negative infinity as the argument approaches zero.
• **Base's Impact**: Different bases, such as 10 (common log) or \(e\) (natural log), alter the function's steepness but not its overall shape.

In this context, the logarithmic function in the exercise works with trigonometric expressions as arguments, making it interesting to consider how these range values apply to the main function. Remember, the function's base fundamentally changes how we perceive growth.

Trigonometric Functions

Trigonometric functions are periodic functions derived from the angles of triangles. They are critical in fields such as physics, engineering, and signal processing. The main functions are sine, cosine, and tangent, each having unique properties. In this exercise, the sine (\(\sin\)) and cosine (\(\cos\)) functions play crucial roles.

• **Periodicity**: Both \(\sin x\) and \(\cos x\) are periodic with a period of \(2\pi\), meaning they repeat every \(2\pi\) units.
• **Amplitude**: The amplitude refers to the wave's height. For \(\sin\) and \(\cos\), the amplitude is constant at 1.
• **Range and Peculiarities**: The range of these functions is \([-1, 1]\), since they oscillate above and below zero endlessly, reflecting typical wave-like motion in mathematics.

When incorporated into a more complex expression, such as \(\sqrt{2}(\sin x - \cos x)\), as in this exercise, it transforms the function's dynamics and changes the resulting output range. Often trigonometric identities are employed to simplify solving problems using these functions.

Euler's Number

Euler's number, represented as \(e\), is akin to the backbone of natural logarithms and exponential functions. Approximately equal to 2.718, it is one of the most important constants in mathematics. This number appears naturally across various phenomena involving growth processes, like compound interest, population growth, and radioactive decay.

• **Exponential Functions**: The function \(e^x\) grows exponentially. It means, with each increase in \(x\), the function rises at a rate proportional to its current value.
• **Properties**: Euler's number features prominently in calculus, particularly when computing derivatives of exponential functions, as these derivatives remain remarkably unchanged.
• **Relation to Logarithms**: \(e\) is the base of natural logarithms, which are used when the change is proportional to the quantity's current size.

In the context of this exercise, exponential expressions use \(e\) to model dynamics that reflect hyperbolic functions like \(\tanh(x)\), which appears in hyperbolic trigonometry. Such functions model realistic growth and decay patterns across various scientific disciplines.