Problem 106 - 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. $$f(x)=\ln \frac{x}{x^{2}+1}$$

Step-by-Step Solution

Answer

The domain of the function is \(x > 0\). It could be observed that the function increases and decreases in the intervals identified from the graph. The approximate relative maximum and minimum values could be identified from the graph.

Step 1: Graph the function

By inputting \(f(x)=\ln \frac{x}{x^{2}+1}\) into the graphing utility, plot the graph. Observe the pattern of the graph to understand how the function behaves with increasing x-values.

Step 2: Find the domain

The domain of the function can be found by identifying for which x-values the function is defined. For any logarithm, the content of the logarithm, or the 'argument', must be greater than zero. Hence, the domain of this function is \(x > 0\). The function is not defined for \(x = 0\) as the argument of the logarithm would be 0 in that case.

Step 3: Identify intervals of increase and decrease

By examining the graph, identify where the graph is increasing and decreasing. The function is increasing when the slope of the graph is positive, and decreasing when the slope is negative. This often translates into the graph 'going upwards' or 'going downwards', respectively.

Step 4: Approximate the relative maximum or minimum values

Identify the high and low points on the graph by inspection. As the problem asks for approximations, providing the approximate x-values of these points should suffice.

Key Concepts

Domain of a Function

Understanding the domain of a function is essential to graphing logarithmic functions properly. The domain refers to all the possible input values (usually represented by 'x') for which the function is defined and will produce a real number output. With logarithmic functions, such as in our exercise \(f(x)=\ln \frac{x}{x^{2}+1}\), the 'argument'—the value inside the logarithm—must be greater than zero. This is because the logarithm of zero or a negative number is not defined in the realm of real numbers.

When we analyze \(f(x)\), we see that the numerator 'x' must be positive to have a positive argument. However, since the denominator \(x^{2}+1\) is always positive for any real number value of 'x', the function will indeed have a positive argument as long as x is positive. Thus, the domain is \(x > 0\). It's crucial to remember that domain restrictions are fundamental to the function's behavior and ensure the function's output is valid and meaningful.

Increasing and Decreasing Intervals

To determine when our logarithmic function is increasing or decreasing, we look at the graph's slope. Intuitively, if the graph of the function is moving 'upwards' as it progresses from left to right, the function is increasing. Conversely, if the graph is moving 'downwards', the function is decreasing.

Graphically, a positive slope indicates an increasing interval, whereas a negative slope indicates a decreasing interval. For our function \(f(x)=\ln \frac{x}{x^{2}+1}\), you'll notice, upon graphing, that some portions of the curve rise while others fall as 'x' increases. These visual changes on the graph indicate the function's behavior on different intervals. By carefully observing the corresponding 'x' values at which the graph transitions from rising to falling or vice versa, we can specify the increasing and decreasing intervals accurately, which helps in understanding the overall shape and the behavior of the function.

Relative Maximum and Minimum Values

Relative maximum and minimum values of a function are points where the function's graph reaches a local 'peak' or 'valley'. These points are particularly important as they indicate extremities in the behavior of the function.

To approximate these values for our logarithmic function \(f(x)=\ln \frac{x}{x^{2}+1}\), we identify the points on the graph where the slope changes from positive to negative (for a maximum) or from negative to positive (for a minimum). Due to their local nature, these points don't necessarily need to be the absolute highest or lowest values the function can achieve, but they are the highest or lowest within their immediate vicinity on the graph.

Once these points are identified by observing the graph, we can approximate their 'x' coordinates. Since logarithmic functions can have complex behaviors, using a graphing utility greatly assists in identifying these points and hence in understanding the function's detailed characteristics.