Problem 102 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$f(x)=\frac{x}{\ln x}$$

Step-by-Step Solution

Answer

The domain of the function is (0, +\infty). The function is decreasing on interval (0,1) and increasing on interval (1, +\infty). There is a relative minimum at x=1 with function value 0, and no relative maximum.

Step 1: Graph the function

Using a graphing utility, plot the given function \(f(x)=\frac{x}{\ln x}\). This will give a visual representation, which will be used to determine the domain, intervals of increase and decrease, and any relative extrema.

Step 2: Determine the domain

By analyzing the function \(f(x)=\frac{x}{\ln x}\), it can be observed that x cannot be equal to 0 and x cannot be less than 0 as the natural logarithm is undefined at x=0 and for negative x. Hence the domain of f is \((0, +\infty)\).

Step 3: Identifying intervals where function is increasing or decreasing

Analyze the graph and look at where the function is rising or falling. With this function, it can be seen that it is decreasing from \((0,1)\), and increasing from \((1, +\infty)\). This means that the function has a minimum at x=1.

Step 4: Approximate any relative maximum or minimum values

Analyze the graph for any visible hills (maximum points) or valleys (minimum points). The function \(f(x)=\frac{x}{\ln x}\) has a minimum at x=1. The function value at x=1 is \(f(1)=\frac{1}{\ln 1} = 0\). There is no relative maximum.

Key Concepts

Domain of a Function

Understanding the domain of a function is a fundamental aspect of algebra that describes the set of all possible input values (usually represented by 'x') for which the function is defined. For the given function \(f(x)=\frac{x}{\ln x}\), determining the domain is crucial because it includes an operation that isn't defined for all real numbers: taking the natural logarithm, denoted as \(\ln x\).

The natural logarithm is only defined for positive real numbers. Thus, for our function, \(x\) must be greater than zero. Moreover, \(x\) cannot be zero because we cannot divide by zero (which would happen if we were to take \(\ln 0\)). Therefore, the domain of the function is all positive real numbers, expressed as \( (0, +\infty) \).

In general, determining the domain of a function involves identifying all input values that do not lead to undefined or forbidden operations, such as division by zero, square roots of negative numbers, or logarithms of non-positive numbers.

Intervals of Increase and Decrease

Investigating the intervals on which a function increases or decreases gives us valuable insight into its behavior. For our function \(f(x)=\frac{x}{\ln x}\), we identify these intervals by analyzing the output values as \(x\) changes. If the output, or \(f(x)\), gets larger as \(x\) gets larger, the function is increasing. If \(f(x)\) gets smaller, the function is decreasing.

To find these intervals without calculus, one can use a graphing utility. The graph of \(f(x)\) indicates that the function is decreasing from \( (0,1) \) as the \(y\)-values decrease with increasing \(x\)-values within this interval. From \( (1, +\infty) \) the function is increasing, meaning \(f(x)\) rises as \(x\) grows.

This information is not just academic; it helps in understanding how the function behaves between its bounds. For instance, considering a business model, an increasing interval might represent a growing profit, while a decreasing interval can signal a need for strategy reassessment.

Relative Maximum and Minimum

Identifying relative maximum and minimum points provides insight into the peaks and troughs of a function's graph, which are key to understanding its overall shape. These are also referred to as local extrema. A relative maximum is a point where the function's value is higher than all other values in the immediate vicinity, while a relative minimum is the opposite.

In the case of our function \(f(x)=\frac{x}{\ln x}\), we observe a minimum but no maximum. At \(x=1\), the function attains its minimum value, which can be calculated to be \(f(1)=\frac{1}{\ln 1}=0\), since \(\ln 1 = 0\). There is no relative maximum observed on the graph produced by the graphing utility.

Such points are crucial when determining optimal points in various real-life situations, such as finding the least cost or maximum profit. Recognizing and calculating these points can be an essential skill in analysis and decision making.

Using Graphing Utilities in Algebra

Graphing utilities, such as graph calculators or computer software, are invaluable tools in algebra. They allow for visualizing complex functions and identifying their characteristics swiftly. When working with a function like \(f(x)=\frac{x}{\ln x}\), graphing utilities can provide immediate visual feedback on its shape and behavior, which is especially useful for learning and confirming algebraic insights.

With graphing utilities, we can swiftly approximate important features like the domain, intervals of increase and decrease, and relative extrema without resorting to more complicated analytical methods. They are particularly useful when the algebraic manipulation of a function to find these characteristics is not straightforward.

In our exercise, by using a graphing utility to plot \(f(x)\), we visually confirmed the domain \( (0, +\infty) \) and easily identified the interval of increase \( (1, +\infty) \) and the relative minimum at \(x=1\). This technology is part of a modern approach to learning algebra, improving accessibility, and allowing for a more intuitive understanding of mathematical concepts.