Problem 53 - University Calculus: Early Transcendentals

Question

What is special about the functions $$f(x)=\sin ^{-1} \frac{x-1}{x+1}, \quad x \geq 0, \quad \text { and } \quad g(x)=2 \tan ^{-1} \sqrt{x} ?$$ Explain.

Step-by-Step Solution

Answer

The functions are inverse trigonometric identities with specific transformations, essentially representing a similar type of transformation for different domains and ranges.

Step 1: Consider the function f(x)

The function is given by \( f(x) = \sin^{-1} \left( \frac{x-1}{x+1} \right) \) for \( x \geq 0 \). Begin by analyzing the expression inside the inverse sine function. Notice that as \( x \to \infty \), \( \frac{x-1}{x+1} \to 1 \), and as \( x \to 0 \), \( \frac{x-1}{x+1} \to -1 \). Therefore, \( f(x) \) ranges from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

Step 2: Evaluate the function g(x)

The function is \( g(x) = 2 \tan^{-1} \sqrt{x} \). Note that \( \tan^{-1} \sqrt{x} \) has a range from 0 to \( \frac{\pi}{2} \) as \( \sqrt{x} \) varies from 0 to \( \infty \). Therefore, \( g(x) \) has a range from 0 to \( \pi \).

Step 3: Compare the ranges

Now, let's compare the ranges of \( f(x) \) and \( g(x) \). It's observed that both \( f(x) \) and \( g(x) \) are functions defined for \( x \geq 0 \). The range of \( f(x) \) is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), while the range of \( g(x) \) is from 0 to \( \pi \).

Step 4: Analyze the identities of inverse functions

A key observation is that the derivatives with respect to \( x \) of \( f(x) \) and \( g(x) \) result in consistently related functions, hinting at a structural similarity aside from their end behavior. Specifically, they both describe the arc functions: \( \sin^{-1} \) for \( f(x) \) and \( \tan^{-1} \) for \( g(x) \). Under appropriate transformations and considering their behaviors, \( f(x) \) and \( g(x) \) are inverse trigonometric representations with specific transformations involved.

Key Concepts

Function Analysis

Understanding the behavior of a function is key in exploring its characteristics and applications. Function analysis involves examining the input-output relationship, determining its range and domain, and identifying any special features.
For instance, consider the function \( f(x) = \sin^{-1} \left( \frac{x-1}{x+1} \right) \). By examining its expression, we find:

• As \( x \) approaches infinity: \( \frac{x-1}{x+1} \) moves toward 1.
• When \( x \) is close to zero: \( \frac{x-1}{x+1} \) tends to -1.

This behavior helps us conclude the function's range is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \).
Similarly, for \( g(x) = 2 \tan^{-1} \sqrt{x} \), we see how \( \tan^{-1} \) shifts as \( \sqrt{x} \) varies, leading to a range from 0 to \( \pi \).
This type of analysis, by observing limits and transformations, uncovers intricate behaviors of functions.

Arc Functions

Arc functions, often known as inverse trigonometric functions, are essential for determining angles when given a trigonometric value. They effectively reverse the trigonometric functions—transforming ratios back into angles.
The arc sine function \( \sin^{-1}(x) \) produces an angle whose sine is \( x \). As seen in \( f(x) = \sin^{-1} \left( \frac{x-1}{x+1} \right) \), this arc function helps discern angle values from varying ratios produced by the function's input.
The arc tangent function \( \tan^{-1}(x) \), used in \( g(x) = 2 \tan^{-1} \sqrt{x} \), similarly identifies angles, essential when dealing with ratios resulting from the square root transformation of \( x \).
Both arc sine and arc tangent serve as building blocks that allow deeper interpretations of geometric relationships, transforming numerical values into meaningful angular interpretations.

Trigonometric Identities

Trigonometric identities form the foundational principles connecting different trigonometric functions, often simplifying complex expressions and calculations. This concept is evident in the transformation methods used in inverse trigonometric functions.
For \( f(x) = \sin^{-1} \left( \frac{x-1}{x+1} \right) \) and \( g(x) = 2 \tan^{-1} \sqrt{x} \), their behavior is intertwined with trigonometric identities that reshape one form into another, revealing structural similarities.
These identities help us understand phase shifts, angle differences, and symmetry properties, making complex calculations more manageable through transformed frameworks.
By applying identities like \( \tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)} \) or properties related to symmetry, the expressions take meaningful forms, aiding in intuitive interpretations and effective problem-solving strategies.