Question

In Exercises \(25-26,\) show that the function is one-to-one, and graph its inverse. $$y=\tan x\( \)-\frac{\pi}{2}\( \)< x <$$\frac{\pi}{2}$$

Short answer

Yes, the function \( y = tan(x), -\frac{\pi}{2} < x < \frac{\pi}{2} \) is one-to-one. Its inverse function is \( x = tan^{-1}(y) \) which increases in the domain \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). The graph of the inverse function is a reflection of the original function about the line y=x, with horizontal asymptotes at \(y=\frac{\pi}{2}\) and \(y=-\frac{\pi}{2}\).

Step-by-step solution

Verify if function is one-to-one

In order to determine if a function is one-to-one, we must ensure that no two different inputs x1 and x2 in the domain produce the same output. For the function \(y = \tan(x)\), this is possible only in between the interval \[-\frac{\pi}{2}, \frac{\pi}{2}\] as stated in the problem, because outside this interval, the tangent function starts to repeat its values. Hence, we can confirm that the function is indeed one-to-one.

Find the inverse of the function

The inverse of a function is a procedure to 'undo' the effect of the function. Thus, the inverse of \(y = \tan(x)\) is \(x = \tan^{-1}(y)\). Thus, the graph of the inverse function will be a reflection of the original function across the line y=x.

Graph the inverse function

To graph the inverse of the function, we simply need to sketch a graph of the tangent function within the interval \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), and then reflect this about the line y=x. We'll find the vertical asymptotes at y=\(\frac{\pi}{2}\) and y=-\(\frac{\pi}{2}\). For the inverse function, these vertical asymptotes become horizontal asymptotes. In the domain \(-\frac{\pi}{2}\) to \(+\frac{\pi}{2}\), the function is increasing and hence, its inverse function is also increasing.

Key concepts

One-to-One Functions

Understanding one-to-one functions is essential for exploring inverse functions. A function is considered to be one-to-one if it never assigns the same value to two different inputs. In other words, each element of the domain is paired with a unique element of the range.

For a function to have an inverse, it must be one-to-one. This ensures that the inverse will also be a function, assigning one output for each input. A quick method to test if a function is one-to-one is by using the horizontal line test. If any horizontal line crosses the function's graph more than once, then the function is not one-to-one.

Exercise Improvement Advice:

When explaining this concept with an exercise, it is valuable to illustrate the horizontal line test. Additionally, discussing how modifications to a function's domain can turn a non-one-to-one function into one that satisfies the criteria can offer deeper insights for students.

Tangent Function

The tangent function, denoted as \( y = \tan(x) \) , is a periodic trigonometric function with a fundamental period of \( \pi \) radians. It is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. However, in the Cartesian plane, \( \tan(x) \) is depicted as a curve that repeats every \( \pi \) radians and has vertical asymptotes at \( x = \frac{\pi}{2} + k\pi \) for any integer \( k \).

Specifically, in the context of the exercise, we are interested in the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) where the tangent function does not repeat its values and is continuously increasing. This property makes \( \tan(x) \) one-to-one and thus invertible over this interval.

Exercise Improvement Advice:

A helpful tip for students might be to point out the symmetrical nature of the tangent function about the origin, explaining its odd symmetry, which can further aid in graphing exercises.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the trigonometric functions with restricted domains. They are used to find the angle that corresponds to a given trigonometric ratio. For the tangent function, its inverse is denoted as \( y = \tan^{-1}(x) \) or \( y = \arctan(x) \).

To graph an inverse trigonometric function such as \( \tan^{-1}(x) \) , you reflect the original trigonometric graph across the line \( y=x \). This reflection changes vertical asymptotes into horizontal ones and vice versa. The graph of \( \tan^{-1}(x) \) increases continuously from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) and it has horizontal asymptotes at these values.

Exercise Improvement Advice:

When dealing with inverse trigonometric functions in exercises, it is crucial to explain the concept of restricted domains so students understand why these functions only operate within certain intervals. By stressing the concept of the 'principal value', which represents the main range of the angles produced by inverse trigonometric functions, learners will gain a clearer understanding of these functions' behaviors and limitations.