Problem 21 - Precalculus with Calculus Previews

Question

In Problems \(21-24\), express the given composition of functions as a rational function of \(x\), where \(x>0\). $$ \sinh (\ln x) $$

Step-by-Step Solution

Answer

The function is \(\frac{x^2 - 1}{2x}\).

Step 1: Recall the Definitions

Recall that the hyperbolic sine function is defined as: \[ \sinh(y) = \frac{e^y - e^{-y}}{2} \]We also know the natural logarithm function: \[ \ln(x) = y \implies e^y = x \rightarrow y = \ln x \]

Step 2: Substitute the Expression

Substitute the expression for \(y\) into the hyperbolic sine function: \[ \sinh(\ln x) = \frac{e^{\ln x} - e^{-\ln x}}{2} \]

Step 3: Simplify Exponential Terms

Utilize the property of logarithms and exponents where \(e^{\ln x} = x\) and \(e^{-\ln x} = \frac{1}{x}\):\[ \sinh(\ln x) = \frac{x - \frac{1}{x}}{2} \]

Step 4: Express as a Rational Function

Combine the terms into a single fraction:\[ \sinh(\ln x) = \frac{x^2 - 1}{2x} \]This is the rational function form.

Key Concepts

Rational Functions

Rational functions are intriguing mathematical expressions that are built using ratios of polynomials. By definition, a rational function is expressed as a fraction where the numerator and the denominator are both polynomials. For example, a simple rational function could look like \( \frac{1}{x} \) or more complex like \( \frac{x^2 - 1}{2x} \).

These functions are valuable because they can describe complex relationships beyond linear or quadratic ones. They often arise in calculus and real-world modeling scenarios.

• The numerator and denominator in a rational function can have any degree, but the denominator should never be zero to avoid undefined values.
• Understanding asymptotes, which are the lines that the graph of the function approaches as \(x\) becomes very large or very small, is important when graphing rational functions.
• In the exercise "\( \sinh(\ln x) = \frac{x^2 - 1}{2x} \)", we see that simplifying hyperbolic and logarithmic expressions can lead us to express them as rational functions.

Exploring these conversions helps in grasping how different mathematical concepts interact.

Natural Logarithms

Natural logarithms are a special kind of logarithm that use the constant \(e\) as their base. Whereas typical logarithms are base 10 or some other numbers, natural logs (\(\ln\)) focus specifically on \(e\), which is approximately equal to 2.718. This base \(e\) is significant because it appears naturally in many mathematical contexts, especially those involving growth or decay processes.

Understanding natural logarithms requires grasping a few key ideas:

• Natural logarithms are the inverse operation of exponential functions, which means \(\ln(e^y) = y\) and conversely, \(e^{\ln x} = x\).
• They provide a way to analyze multiplicative effects as additive, making complex calculations more manageable.
• In problems where one is asked to evaluate expressions like \(\sinh(\ln x)\), recognizing the role of the natural logarithm is crucial for substitution and simplification steps.

Natural logarithms thus bridge various mathematical operations, making them a cornerstone in higher mathematics.

Exponential Functions

Exponential functions are fundamental to understanding growth and decay in mathematics and the natural world. Any function of the form \( f(x) = a \cdot e^{bx} \) where \(e\) is the base of the natural logarithm, represents exponential growth or decay.

Let's delve into some essential features of exponential functions:

• They grow rapidly, so small changes in \(x\) can lead to large changes in \(f(x)\). For instance, financial models and population growth often use exponential functions due to their ability to model compounding effects.
• Exponents can be manipulated using properties such as \(e^{a+b} = e^a \cdot e^b\), which help in simplifying expressions in calculus and algebra.
• In the particular expression \(\sinh(\ln x)\), exponential functions surface through the computation \(e^{\ln x} = x\) and \(e^{-\ln x} = \frac{1}{x}\), showcasing the inverse relationship between exponential and logarithmic functions.

This understanding is crucial for working through mathematical problems that involve growth patterns or transformations, as well as understanding calculus concepts like derivatives and integrals of exponential functions.