Question

Explain why domain restrictions are necessary to create inverse trigonometric functions rather than relations.(4 points)

Answer 1

Inverse trigonometric functions are derived from the original trigonometric functions, which are periodic and hence do not pass the horizontal line test. To create a proper inverse function, certain domain restrictions are necessary for the following reasons:

1. **One-to-One Property**: Trigonometric functions are not one-to-one over their entire domains because they repeat their values. For instance, the sine function, \( \sin(x) \), has the same output for multiple input angles (e.g., \( \sin(0) = \sin(\pi) = \sin(2\pi) = 0 \)). By restricting their domains (e.g., defining the sine function only on the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\)), we ensure that each input corresponds to exactly one output, which is a requirement for a function to have an inverse.

2. **Existence of Unique Outputs**: An inverse function needs to assign a unique output for each input. Without domain restrictions, inverse relations would yield multiple outputs for a single input due to the periodic nature of trigonometric functions. For example, if we take \( \sin(x) \) without restrictions, \( \sin^{-1}(0) \) would not be uniquely defined because it corresponds to \( x = 0, \pi, 2\pi, \ldots \). By limiting the domain, we ensure that each value in the range of the original function maps back to a unique value in the domain of the inverse function.

3. **Defined Range**: Inverse trigonometric functions need to have specific ranges that match the outputs of the restricted domains. For example, \( \arcsin(x) \) is defined on the range \([-1, 1]\) and produces outputs in \([- \frac{\pi}{2}, \frac{\pi}{2}]\). This range ensures that the inverse function remains continuous and adequately represents the angles corresponding to the values of the sine function.

4. **Graphical Representation**: When we graph a function and its inverse, the inverse should be a reflection across the line \(y = x\). Domain restrictions ensure that the graph of the inverse function does not become disjoint or overlapping, which would make it a relation rather than a function. Restricting the domain of original trigonometric functions allows their inverses to be well-behaved and easily interpretable geometrically.

In summary, domain restrictions are essential for trigonometric functions to be transformed into inverse functions that are well-defined, one-to-one, uniquely related, and easily representable.