Problem 83 - Precalculus with Calculus Previews

Question

In Problems 83 and 84 , give the domain of the function \(f\). Find all zeros of the function of \(f\). Use a graphing utility to obtain the graph of \(f\). $$ f(x)=\sin (\ln x) $$

Step-by-Step Solution

Answer

Domain: (0, ∞), Zeros: x = e^{nπ} for integers n.

Step 1: Identify the Domain Constraints

To determine the domain of \(f(x) = \sin(\ln x)\), we need to look at where the function inside the sine function is defined. The natural logarithm \(\ln(x)\) is only defined for \(x > 0\). Therefore, the domain of \(f(x)\) is \((0, \infty)\).

Step 2: Find the Zeros of the Function

To find the zeros of the function, we need \(\sin(\ln x) = 0\). The sine function equals zero at integer multiples of \(\pi\), i.e., \(\ln x = n\pi\) where \(n\) is an integer. Thus, \(x = e^{n\pi}\). Therefore, the zeros of \(f(x)\) are at \(x = e^{n\pi}\), for integers \(n\).

Step 3: Graphing the Function

Using a graphing utility, plot \(f(x) = \sin(\ln x)\). The graph will oscillate between -1 and 1, mirroring the behavior of the sine function applied over the positive real numbers as input.

Key Concepts

Domain of a Function

The domain of a function is essentially the set of all possible input values (or "x-values") for which the function is defined. For the function \(f(x) = \sin(\ln x)\), we need to determine where \(\ln(x)\) is valid, as the natural logarithm has specific constraints. The natural logarithm \(\ln x\) is defined only for positive values of \(x\). As such, \(x\) must be greater than zero to avoid undefined expressions.
This leads us to conclude that the domain of the function is all positive numbers, expressed in interval notation as \((0, \infty)\). This means you can input any positive number into the function, and the result will be defined.
Understanding the domain is crucial as it helps avoid mathematical errors in calculations and ensures the function's application in real-life problems is accurate.

Natural Logarithm

Natural logarithms are logarithms to the base \(e\), where \(e \approx 2.718\). It's a widely occurring mathematical constant that appears in various natural processes and phenomena. The function \(\ln(x)\) represents the power to which we need to raise \(e\) to get \(x\).
The notable characteristic of the natural logarithm is its definition only for positive \(x\). This is because logarithms are fundamentally based on exponents, and exponentials of a positive base cannot equal zero or negative numbers. Thus, the natural logarithm forms the core part of the domain consideration for \(f(x) = \sin(\ln x)\).
Knowledge of \(\ln(x)\) is not only crucial for studying growth processes, decay, and oscillations in math but also extends to fields such as physics, biology, and economics.

Sine Function

The sine function is a periodic trigonometric function that oscillates between -1 and 1. It is defined for all real numbers and is usually associated with angles and waves. In the context of the function \(f(x) = \sin(\ln x)\), the sine function takes the natural logarithm's output, which compresses its input to the positive real numbers.
Periodicity is a key feature of sine, where \(\sin(x)\) repeats every \(2\pi\). This property crucially impacts the graph of \(f(x)\), as it means \(f(x)\) will also show repeating patterns, though along the stretched logarithmic scale.

• The sine function equals zero at every integer multiple of \(\pi\), making it easy to spot where \(f(x) = \sin(\ln x)\) will also reach zero, provided \(\ln(x) = n\pi\).

Recognizing the behavior of the sine function will enhance your understanding of wave-like patterns and oscillatory motions in different applications.

Zeros of a Function

The zeros of a function are the input values where the function's output is zero, essentially the "roots" or "solutions" of the function's equation. For the function \(f(x) = \sin(\ln x)\), finding the zeros involves setting the function equal to zero and solving for \(x\).
The sine function has zeros at integer multiples of \(\pi\), specifically when the argument equals \(0, \pm\pi, \pm2\pi, \ldots\). Thus, for \(\sin(\ln x) = 0\), we need \(\ln x = n\pi\), where \(n\) is an integer.
To solve for \(x\), exponentiate both sides of the equation, yielding \(x = e^{n\pi}\). This means \(x\) will take forms like \(e^\pi, e^{2\pi}, e^{3\pi}\), and so forth, depending on the integer \(n\).

• Understanding where the function touches the x-axis (the zeros) is essential for graph analysis and signal processing.

Finding the zeros offers insight into the function's behavior, which helps in multiple scientific and engineering applications.