Problem 14 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

The inverse of function \(y=\frac{10^{x}-10^{-x}}{10^{x}+10^{-x}}\) (a) \(y=\frac{1}{2} \log _{10}\left(\frac{1+x}{1-x}\right)\) (b) \(\log _{10} \frac{1-x}{1+x}\) (c) \(2 \log \frac{1+x}{1-x}\) (d) None of these

Step-by-Step Solution

Answer

Correct answer is (a): \( y=\frac{1}{2} \log _{10}\left(\frac{1+x}{1-x}\right) \).

Step 1: Understand the Original Function

The original function is given as \( y = \frac{10^x - 10^{-x}}{10^x + 10^{-x}} \). This can be rewritten in terms of hyperbolic functions as \( y = \tanh(a x) \), where \( a = \log_{10}(e) \), recognizing a hyperbolic tangent form.

Step 2: Find Inverse Expression for the Function

Knowing the function is in the form of \( y = \tanh(a x) \), the inverse is \( x = \text{artanh}(y) \). Expressing \( \text{artanh}(y) \) in terms of logarithms, we find \( x = \frac{1}{2} \log \left( \frac{1+y}{1-y} \right) \). To convert this into base 10: \( x = \frac{1}{2} \log_{10} \left( \frac{1+y}{1-y} \right) / \log_{10}(e) \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogues of trigonometric functions, but they relate to the hyperbola, rather than the circle. In mathematics, these functions include the hyperbolic sine \( \sinh(x) \), hyperbolic cosine \( \cosh(x) \), and the hyperbolic tangent \( \tanh(x) \). These functions, though similar to their trigonometric counterparts, deal with exponential growth and decay—a concept frequently used in calculus and engineering.
The hyperbolic tangent function, denoted as \( \tanh(x) \), is specifically pivotal because it maps real numbers to the interval \( (-1, 1) \). It is defined by the formula \( \tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \).
Just like trigonometric identities, hyperbolic identities assist in rewriting expressions, making them valuable in solving equations related to hyperbolas.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. When we talk about the logarithm of a number, we refer to the exponent by which a base number is raised to yield that number. The general form is \( \log_b(x) = y \) implies \( b^y = x \).
Base 10 logarithms, or common logarithms, are especially prevalent in scientific calculations, often abbreviated to \( \log_{10} \) or simply \( \log \). Logarithmic functions reduce multiplicative processes to additive ones and significantly simplify the operations on very large or very small numbers.
In mathematical contexts, particularly calculus, logarithmic functions play a crucial role due to their properties of derivative and integration, making them essential for understanding rates of change and growth processes.

Tanh Function

The \( \tanh(x) \) function is one of the core hyperbolic functions. It specifically represents the hyperbolic tangent, used to model scenarios that involve exponential spreads or exponential convergence. In formulas, \( \tanh(x) \) can be expressed as \( \tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \) and is closely related to the identity and exponential functions.
The inverse of the hyperbolic tangent function is called the area hyperbolic tangent or artanh, symbolized as \( \text{artanh}(x) \). The inverse function is vital for scenarios where computing initial inputs based purely on outputs is necessary. It is calculated using the formula \( \text{artanh}(x) = \frac{1}{2} \log \left(\frac{1+x}{1-x} \right) \), making it evident how hyperbolic functions are intertwined deeply with logarithmic functions.
Understanding these relationships aids in tackling complex problems involving hyperbolic properties, like those encountered in hyperbolic geometry and signal processing.