Problem 15 - Calculus of a Single Variable

Question

Use a graphing utility to graph the function. Then use the Horizontal Line Test to determine whether the function is one-to-one on its entire domain and therefore has an inverse function. \(f(\theta)=\sin \theta\)

Step-by-Step Solution

Answer

The function \(f(\theta) = \sin \theta\) is not one-to-one on its entire domain and does not have an inverse.

Step 1: Graph the Function

Using a graphing utility, plot the graph for the function \(f(\theta) = \sin \theta\). The function will display a wave-like pattern, which is a typical characteristic of the sine function.

Step 2: Use the Horizontal Line Test

Now, use the horizontal line test. This test states that if any horizontal line crosses the graph of the function more than once, the function does not have an inverse. Apply this line on the graph. It will intersect the graph multiple times because of the periodic nature of the sine wave.

Step 3: Determine whether the function has an inverse

Since the graph intersects at multiple points, we can conclude that the function is not one-to-one on its entire domain because it doesn't pass the horizontal line test. Hence, the function \(f(\theta) = \sin \theta\) does not have an inverse function on its entire domain.

Key Concepts

Horizontal Line Test

The Horizontal Line Test is a useful method to determine if a function is one-to-one. When a function is one-to-one, it means that each element of the range corresponds to exactly one element of the domain. This property is essential for a function to have an inverse.
To apply the Horizontal Line Test, you draw horizontal lines across the graph of the function. If any horizontal line crosses the graph more than once, then the function fails the test and is not one-to-one.
For instance, the graph of the sine function, which oscillates between 1 and -1, clearly fails this test because horizontal lines will intersect the curve at multiple points. This reveals that the sine function does not have an inverse that exists over its entire domain. However, by restricting the domain of the sine function, we can create an inverse on that specific interval. It's crucial to remember that the entire domain must pass the test for the inverse to exist without restrictions.

Periodic Functions

A periodic function is one that repeats its values at regular intervals or periods. Some common examples of periodic functions include sine and cosine functions, which are fundamental in trigonometry.
The sine function, for instance, has a period of \(2\pi\), meaning that every \(2\pi\) units, the wave pattern repeats itself. This characteristic leads to functions like sine having a wavelike shape.
Periodic functions usually don't have inverses over their entire domains because the values repeat at regular intervals. When applying the Horizontal Line Test, this repeatability is apparent as horizontal lines can intersect the wave at multiple points within one period. To resolve this, domains of periodic functions are often restricted so that they become one-to-one, allowing for an inverse to be created on those specific intervals.
Understanding periodicity is essential in recognizing when and how inverses of trigonometric functions can be described and utilized.

Graphing Utilities

Graphing utilities are incredibly helpful tools in visualizing mathematical functions and relationships. These tools include graphing calculators, computer software, and online graphing applications that allow students and mathematicians to plot functions.
By using graphing utilities, one can easily assess the visual behavior of a function, identify patterns, and apply tests like the Horizontal Line Test. For periodic functions such as the sine function, graphing utilities display their oscillating nature and help in understanding how certain restrictions can create inverse functions.
These utilities offer a hands-on approach to learning, as one can adjust parameters and instantly see their effects, fostering a deeper understanding of the mathematical concepts being studied. Graphing utilities are especially useful when dealing with complex functions that are difficult to visualize purely through their algebraic expressions. They make abstract concepts more concrete and accessible to learners at all levels.