Problem 73 - Algebra and Trigonometry

Question

a. Graph the restricted secant function, \(y=\sec x,\) by restricting \(x\) to the intervals \(\left[0, \frac{\pi}{2}\right)\) and \(\left(\frac{\pi}{2}, \pi\right]\) b. Use the horizontal line test to explain why the restricted secant function has an inverse function. c. Use the graph of the restricted secant function to graph \(y=\sec ^{-1} x\).

Step-by-Step Solution

Answer

a. We have graphed the secant function within the restricted intervals \(\left[0, \frac{\pi}{2}\right)\) and \(\left(\frac{\pi}{2}, \pi\right]\). b. The Horizontal Line Test shows that the restricted secant function has an inverse. c. The graph of the inverse secant function \(\sec^{-1}(x)\) is obtained by reflecting the graph of the restricted secant function across the line \(y=x\).

Step 1: Graphing restricted Secant Function

First, recall that the secant function, \(\sec(x)\), is the reciprocal of the cosine function. Hence, it's undefined wherever cosine is zero. The cosine function is zero at \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\). Since we are focusing on the intervals \(\left[0, \frac{\pi}{2}\right)\) and \(\left(\frac{\pi}{2}, \pi\right]\), eliminate the undefined points. Thus, you get two separate parts of the secant function's graph, showing vertical asymptotes at \(x=\frac{\pi}{2}\) within the intervals of interest.

Step 2: Horizontal Line Test

To discuss the invertibility of a function, the Horizontal Line Test (HLT) is used. The HLT states that if any horizontal line intersects the graph of the function in at most one point, the function has an inverse. Now our secant function graph from step 1 has passed this test within the given restricted 'x' intervals. This means that this restricted secant function has an inverse.

Step 3: Graph of Inverse Secant Function

Now remember that the graph of an inverse function is a reflection of the original function's graph about the line \(y=x\). Thus for our restricted secant function \(\sec^{-1}(x)\), we’ll reflect the graph of the restricted secant function across the line \(y=x\). Note that the range of the inverse secant function will be within the original restricted 'x' intervals, i.e, \(\left[0, \frac{\pi}{2}\right)\) and \(\left(\frac{\pi}{2}, \pi\right]\).

Key Concepts

Horizontal Line Test

The Horizontal Line Test (HLT) is a simple way to determine if a function has an inverse. When you apply the horizontal line test to a graph, you are checking if any horizontal line crosses the graph more than once. If it doesn't, the function can be inverted. For the secant function, we use a restricted domain to ensure that any horizontal line intersects the function at most once.

When the secant function is restricted to the intervals \([0, \frac{\pi}{2})\) and \((\frac{\pi}{2}, \pi]\), it becomes one-to-one. This makes it possible to find an inverse function. So, performing the HLT on the restricted secant function shows that it can become the inverse secant function, \(\sec^{-1}(x)\).

To summarize:

• Confirm that horizontal lines intersect at only one point within the restricted domain.
• This confirms the existence of an inverse function.

Graphing Trigonometric Functions

Graphing trigonometric functions like the secant can initially seem complex, but breaking it down helps. Remember, the secant function is defined as the reciprocal of the cosine function, \(\sec(x) = \frac{1}{\cos(x)}\). This dependency creates vertical asymptotes wherever the cosine function is zero.

To graph the restricted secant function:

• Identify and plot the vertical asymptotes at points where \(\cos(x) = 0\), such as \(x = \frac{\pi}{2}\).
• Focus on the intervals \([0, \frac{\pi}{2})\) and \((\frac{\pi}{2}, \pi]\), ignoring the undefined points.
• Draw the curves of \(\sec(x)\) in these intervals. Each "piece" of the graph mirrors the behavior of the cosine function but is flipped and extended vertically.

Graphing these sections clearly helps visualize both the behavior and the discontinuities of the secant function.

Restricted Domain

A restricted domain is necessary to make some trigonometric functions one-to-one, and thus invertible. With the secant function, its periodicity means that without restriction, it wouldn't pass the horizontal line test because it repeats its values infinitely.

By focusing only on the segments \([0, \frac{\pi}{2})\) and \((\frac{\pi}{2}, \pi]\), we avoid the points where the function is undefined, and more importantly, ensure no horizontal line crosses more than one part of the graph.

Benefits of using a restricted domain:

• The function becomes continuous and one-to-one within the specified intervals.
• This allows for a well-defined inverse function, \(\sec^{-1}(x)\).
• Graphically, it simplifies complex periodic behavior into manageable sections.

Restricting the domain is a common practice in calculus that helps to deal with the complexities of periodic trigonometric functions.