Problem 4 - A Textbook of Differential Calculus

Question

If \(f(x)\) is continuous such that \(|f(x)| \leq 1, \forall x \in R\) and \(g(x)=\frac{e^{f(x)}-e^{|f(x)|}}{e^{f(x)}+e^{|f(x)|}}\), then range of \(g(x)\) is (a) \([0,1]\) (b) \(\left[0, \frac{e^{2}+1}{e^{2}-1}\right]\) (c) \(\left[0, \frac{e^{2}-1}{e^{2}+1}\right]\) (d) \(\left[\frac{1-e^{2}}{1+e^{2}}, 0\right]\)

Step-by-Step Solution

Answer

The range of \( g(x) \) is \( \left[ \frac{1-e^{2}}{1+e^{2}}, 0 \right] \).

Step 1: Understand the range of f(x)

The function \(|f(x)| \leq 1\) suggests that the values of \(f(x)\) are within the interval \([-1, 1]\). This forms the domain over which we need to analyze \(g(x)\).

Step 2: Substitute boundary values into g(x)

We need to analyze \(g(x)\) for boundary values \(f(x) = -1\) and \(f(x) = 1\) due to their potential to yield extreme function values. For \(f(x) = 1\), we get \(g(1) = \frac{e^{1} - e^{1}}{e^{1} + e^{1}} = 0\). For \(f(x) = -1\), compute \(g(-1) = \frac{e^{-1} - e^{1}}{e^{-1} + e^{1}} = \frac{1/e - e}{1/e + e}\).

Step 3: Simplify g(-1) for the range

Calculate \(g(-1) = \frac{1/e - e}{1/e + e} = \frac{1 - e^2}{1 + e^2}\). This value represents the lower bound of \(g(x)\), while the upper bound is defined by \(g(1)=0\). Hence, the range of \(g(x)\) includes values between \(\frac{1 - e^2}{1 + e^2}\) and \(0\).

Step 4: Analyze the behavior of exponential terms

Since \(|f(x)| \leq 1\), \(f(x)\) and \(|f(x)|\) can both be -1, 0, or 1. The function \(g(x)\) effectively calculates the limit behavior: at \(f(x) = 0, g(x)\) results in 0; when \(f(x)\) swings to its extremes, \(-1 \) or \(1\), the function takes on values delimiting \(\frac{1 - e^2}{1 + e^2}\) to 0.

Key Concepts

Range of Functions

The range of a function describes all possible output values it can produce, often seen as the "y-values" on a graph. To determine this range, we explore how the function behaves across its domain.

• In our exercise, the function is given as \(g(x)\).
• The task was to identify the range of \(g(x)\), knowing it depends on \(f(x)\) being continuous and bounded by \([-1, 1]\).
• We substitute extreme values into \(g(x)\) to find its upper and lower bounds.

The idea is that by knowing how \(f(x)\) behaves at these boundary points, we can deduce the full range for \(g(x)\). For this exercise, the range was found by plugging in \( f(x) = -1 \) and \( f(x) = 1 \). This involved calculating specific limits, leading to the conclusion that the range is \( \left[ \frac{1-e^{2}}{1+e^{2}}, 0 \right] \).
Understanding ranges is a crucial aspect of working with functions, as it informs us of all possible outputs and thus, potential solutions.

Exponential Functions

Exponential functions are characterized by their constant growth rate, which is what makes them so intriguing. They have the general form \(a^{x}\), where the base \(a\) is a constant and the exponent \(x\) is the variable.

• In the exercise, exponential functions \(e^{f(x)}\) and \(e^{|f(x)|}\) were used to derive \(g(x)\).
• The exponential nature of these functions is central to how they transform values from \(f(x)\) into the numerator and denominator of \(g(x)\).

The reason exponential functions are pivotal here is due to their impact on growth patterns and limits. When \(f(x)\) equals its upper or lower bounds, the function \(g(x)\) reveals values deeply connected to these exponential transformations.
This highlights an essential aspect of exponential functions: their potential to magnify small changes in \(f(x)\) because of their inherent nature to grow rapidly.

Boundaries of Functions

The boundaries of a function are crucial in understanding its behavior, especially when analyzed for continuity and absolute limits.

• Our exercise discusses the boundaries of \(f(x)\) given by \(|f(x)| \leq 1\).
• This indicates that \(f(x)\) stays within \([-1, 1]\), directing our analysis of \(g(x)\).

Boundaries determine how far a function can go — literally and figuratively. By testing \(g(x)\) at these edges, where \(f(x)\) is at \(-1\) and \(1\), we get a picture of what \(g(x)\) does in its entirety.
Thus, by understanding the constraints on \(f(x)\), we make more informed assessments of the boundaries and eventual range of \(g(x)\). These boundaries not only illustrate the limits but also guide us through the subtle variations within the domain, thereby defining the scope of the function's output.