Problem 66 - Precalculus: A Concise Course

Question

Consider the functions given by \(f(x)=\tan \frac{\pi x}{2}\) and \(g(x)=\frac{1}{2} \sec \frac{\pi x}{2}\) on the interval (-1,1) (a) Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. (b) Approximate the interval in which \(f

Step-by-Step Solution

Answer

The intervals in which \(f(x) < g(x)\) and \(2f(x) < 2g(x)\) are the same. Multiplying both functions by the same number does not change the interval as the inequality relationship between the two functions remains the same.

Step 1: Graphing the functions

Using a graphing utility, graph the two functions \(f(x)=\tan \frac{\pi x}{2}\) and \(g(x)=\frac{1}{2} \sec \frac{\pi x}{2}\) on the same viewing window on the interval (-1,1).

Step 2: Compare the functions

Compare the graphs of the two functions in the interval (-1, 1). By visually inspecting the graphs, approximate the interval in which the function \(f(x) < g(x)\).

Step 3: Double the functions and compare

Now, double both functions and compare them. That is, find the interval in which \(2f(x) < 2g(x)\). Notice if doubling the functions has any effect on the interval.

Step 4: Analysis

Analyse the result you obtained in Step 3. How does the result compare with that of Step 2? You would notice that multiplying both functions by the same number (in this case, 2) does not change the interval since the inequality relationship between the two functions remains the same.

Key Concepts

Graphing Utility

Graphing utilities are powerful tools that help visualize mathematical functions, especially trigonometric ones that can be complex and difficult to interpret through equations alone. When graphing the functions \(f(x)=\tan \frac{\pi x}{2}\) and \(g(x)=\frac{1}{2} \sec \frac{\pi x}{2}\) within the interval (-1, 1), we can effectively use tools such as graphing calculators or software like Desmos or GeoGebra. This allows us to place both functions in the same viewing window.

Seeing the graphs in unison offers a clear depiction of how these functions behave across the given range.

• The tangent function \(\tan \frac{\pi x}{2}\) has vertical asymptotes, causing the graph to stretch infinitely at certain points within the interval, providing crucial insights into its periodic nature.
• The secant function \(\frac{1}{2} \sec \frac{\pi x}{2}\), which is the reciprocal of cosine, introduces different behavior including its own set of asymptotes.

By graphing them together, you can visually analyze their intersections and regions where one function is greater or less than the other. This graphical insight supports better understanding and comparison compared to numerical approaches alone.

Inequality Analysis

Inequality analysis involves determining where one function falls below or exceeds another within a specific interval. In this exercise:

• We seek to identify where the function \(f(x)=\tan \frac{\pi x}{2}\) is less than \(g(x)=\frac{1}{2} \sec \frac{\pi x}{2}\) over the interval (-1, 1).

Using a graphing utility, as discussed, visual inspection can lead to approximations of intervals where the inequality holds true.

You might observe that despite both functions having asymptotic behaviors, their paths intersect each other at certain points. By carefully studying these points, you can approximate the range where \(f(x) < g(x)\).

For a deeper analysis:

• Observe the behavior of the functions near their asymptotes and within segments of stability where they are closer to zero.
• Understanding the derivative or slope changes can also help pinpoint these intervals.

This facilitates a robust comprehension of where exactly one function dominates over the other.

Function Comparison

Function comparison is pivotal in understanding how differences between functions manifest over a domain. Here, in part (c) of the exercise, we explore the idea of doubling both functions:\(2f(x)\) versus \(2g(x)\).

By following the solution's method, we learn that multiplying both sides of an inequality \(f(x) < g(x)\) by a positive constant (2 in this case) retains the inequality's properties. Thus, \(2f(x) < 2g(x)\) will hold true over the same interval as \(f(x) < g(x)\) without any modification in the range.

This demonstrates an essential property of inequalities:

• When both sides of an inequality are uniformly scaled, their relational inequalities remain unchanged.
• This is valuable when scaling equations to match real-world contexts where relative comparison rather than absolute values is key.

Therefore, by comparing these functions scaled, we reinforce our understanding of relative magnitudes and asymptotic behaviors, providing a clearer picture of the functions' interactions across the interval (-1, 1).