Problem 105 - 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. $$f(x)=\ln \frac{x^{2}}{10}$$

Step-by-Step Solution

Answer

The function \(f(x) = \ln \frac{x^{2}}{10}\) can be graphed using a graphing utility and has a domain of \(x \in (-\infty, 0) \cup (0, \infty)\). It is increasing on the interval \((0, \infty)\) and decreasing on \((-\infty, 0)\). There are no relative maximum or minimum values within the domain.

Step 1: Function Graphing

Begin by graphing the function \(f(x) = \ln \frac{x^{2}}{10}\) using a graphing utility. Unfortunately, this can't be specifically demonstrated in text, but various online tools can be used. Make sure to observe form and behavior of the graph.

Step 2: Determining the Domain

The domain of a logarithmic function is all the x-values for which the function is defined. Given that we can't take logarithms of negative numbers or zero, the function \(\ln \frac{x^{2}}{10}\) is defined for \(x \neq 0\). Thus, the domain is \(x \in (-\infty, 0) \cup (0, \infty)\).

Step 3: Increasing and Decreasing Intervals

By examining the graph, it can be observed that the function \(f(x) = \ln \frac{x^{2}}{10}\) is increasing on \((0, \infty)\) and decreasing on \((-\infty, 0)\). It is important to note that with logarithmic functions, the sign of x affects increase/decrease intervals.

Step 4: Relative Maximum or Minimum Values

From the graph, it can be observed that the function does not have any relative maximum. It reaches a minimum at x = 0, but this point isn't included in the domain. So there are no relative maximum or minimum values within the domain.

Key Concepts

Graphing Utility

Graphing utilities are valuable aids when approaching problems involving functions, especially logarithmic ones. For the function f(x) = \( \ln \frac{x^{2}}{10} \), using a graphing utility allows students to visualize the shape and behavior of the log function. Since the actual graph can't be shown in plain text, students are advised to utilize online graphers or graphing calculators. When graphing, students should pay attention to the curve's symmetry regarding the y-axis and how it increases or decreases as it moves away from the origin.

Understanding the visual depiction of the function can offer insights into the function's nature and key characteristics such as domain, range, and intervals of increase or decrease. Encourage students to explore the graphing utility's features, such as tracing the graph to understand its slope and curvature intuitively.

Function Domain

The function domain encompasses all the permissible x-values that a function can accept. With the given function f(x) = \( \ln \frac{x^{2}}{10} \), discerning the domain is critical because logarithms are undefined for non-positive arguments. Hence, the values of x must be such that \(\frac{x^{2}}{10}\) is positive.

Since squaring any real number yields a positive result, the domain for our logarithmic function explicitly excludes zero, leading to the established domain x \( \in (-\infty, 0) \cup (0, \infty)\). It's essential to enforce the understanding that while x=0 gives a zero inside the logarithm, which is not acceptable, all other non-zero values produce a positive outcome suitable for taking a logarithm.

Increasing and Decreasing Intervals

A function's increasing or decreasing intervals can be determined by analyzing its graph. For our logarithmic function f(x) = \( \ln \frac{x^{2}}{10} \), the graphing utility reveals pertinent trends in its behavior. The function is increasing on the interval (0, \infty), indicating that as x moves away from 0 in the positive direction, f(x) grows. Conversely, it's decreasing on the interval (-\infty, 0), suggesting that as x moves towards zero from the negative side, f(x) becomes smaller.

To assist students with the concept, emphasize the nature of logarithmic functions: their rate of increase slows as x becomes larger in positive value. Also, the opposite is true as x approaches zero from the negative side, accelerating the function's decease.

Relative Maximum and Minimum Values

Identifying a function’s relative maximum and minimum values involves finding local highs and lows on its graph. For the function f(x) = \( \ln \frac{x^{2}}{10} \), a graph can reveal that it does not exhibit any relative maximum since the function continues to increase indefinitely as x grows positively. Importantly, while the function touches its lowest point at x=0, this point is excluded from the domain thus no relative minimum exists within the permissible range of x values.

When teaching this concept, clarify the distinction between 'absolute' and 'relative' extrema. An absolute minimum or maximum refers to the lowest or highest value a function achieves over its entire domain, whereas a relative extremum is one that appears within a certain interval – a high or low in the local neighborhood of the graph.