Problem 45 - A Textbook of Differential Calculus

Question

Let \(f: R \rightarrow R\) be a function such that \(f(x)=\frac{e^{|x|}-e^{-x}}{e^{x}+e^{-x}}\). Statement I \(f(x)\) is into function. Statement II \(f(x)\) is many-one function, and the many-one function is not onto.

Step-by-Step Solution

Answer

The function is many-one and not onto.

Step 1: Simplify the Expression for f(x)

First, let's simplify the given function \(f(x) = \frac{e^{|x|}-e^{-x}}{e^{x}+e^{-x}}\). For \(x \geq 0\), \(|x| = x\) and for \(x < 0\), \(|x| = -x\). We need to consider these two cases separately: \(f(x) = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\) when \(x \geq 0\), and \(f(x) = \frac{e^{-x} - e^{-x}}{e^{x} + e^{-x}} = \frac{0}{e^{x} + e^{-x}} = 0\) when \(x < 0\).

Step 2: Evaluate f(x) for x ≥ 0

For \(x \geq 0\), \(f(x) = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} = \tanh(x)\). \(\tanh(x)\) is a function that maps \(\mathbb{R}_{0}^{+}\) to \([0, 1)\), thus not covering the full range of \(\mathbb{R}\), proving it's into.

Step 3: Evaluate f(x) for x < 0

For \(x < 0\), we found \(f(x) = 0\), which means for all negative \(x\), the function value is \(0\). This constant value makes the function many-one.

Step 4: Determine if f(x) is Many-One or Onto

Since for \(x < 0\), the function value is always \(0\), and for \(x \geq 0\), the function value varies within \([0, 1)\), the function is many-one because multiple inputs give the same output (e.g., all negative \(x\) map to \(0\)). Additionally, since its range does not cover all real numbers, it is not onto.

Key Concepts

Function Analysis

Function analysis is about understanding the behavior of a function. In this case, let us consider the function \( f(x) = \frac{e^{|x|} - e^{-x}}{e^{x} + e^{-x}} \). The key here is to simplify \( f(x) \) based on the value of \( x \). For \( x \geq 0 \), \( f(x) \) simplifies to \( \tanh(x) \), representing a hyperbolic tangent function. This results in any positive x having potentially different outputs corresponding to values in the range [0, 1).
For \( x < 0 \), the absolute value \(|x|\) turns into \(-x\), resulting in \( f(x) = 0 \). Thus, any negative input yields a constant output of 0, revealing aspects of distribution and range behavior specific to input conditions.

Piecewise Functions

Piecewise functions are functions defined by different expressions over different intervals. In our context, the function \( f(x) \) exhibits different behavior depending on whether \( x \) is negative or non-negative.

• For \( x \geq 0 \), \( f(x) = \tanh(x) \). This makes the part of the function defined for positive and zero inputs behave like the hyperbolic tangent function, which is continuous and increasing.
• For \( x < 0 \), \( f(x) = 0 \). Here, the function is constant for all negative inputs, resulting in a flat line at zero.

Such a division characterizes the function as piecewise, crucial in analyzing how the function behaves differently across its domain.

Onto Functions

An onto function, or surjective function, means every element in the codomain is mapped from at least one element in the domain. In the case of our function \( f(x) \), the range is limited to values between 0 and 1 for non-negative inputs \((x \geq 0)\), and is exactly 0 for negative inputs \((x < 0)\).
This restriction effectively keeps the function from being onto with respect to the entire set of real numbers \( \mathbb{R} \) because the entire real line isn't covered by the function outputs. If a function does not have every value in the codomain mapped to it, it cannot be considered onto.

Many-One Functions

A function is called many-one if two or more elements from the domain map to the same element in the codomain. In this case, for all negative \( x \), the function \( f(x) \) maps to 0, indicating that it is many-one for this interval. This feature contrasts with one-to-one functions, where different inputs must map to different outputs.
Another aspect is the range of the function for non-negative \( x \), where each positive and zero \( x \) potentially yields a unique output unless the value is exactly zero. Nonetheless, the presence of multiple negative inputs resulting in a single output, zero, emphasizes the many-one nature of this function. It's important to note this many-one characteristic contributes to the function not being onto because not every possible output is achieved.