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Chapter 4: Problem 105

Describe the restriction on the tangent function so that it has an inverse function.

Short Answer

Expert verified
The restriction on the tangent function so that it has an inverse is \(-\pi/2 < x < \pi/2\).

Step by step solution

01

Understanding tangent function

Tangent function is a periodic function with period \( \pi \), repeating its pattern of variations over every interval of \( \pi \). Making the tangent function one-to-one on the entire number line will require restricting it to a specific interval over which it captures a single pattern.
02

Setting the restriction

A typical choice for restricting the domain of the tangent function is the open interval \(-\pi/2, \pi/2\), exclusive. This is because tangent is undefined (the function has vertical asymptotes) at these two points.
03

Proof of one-to-one function and inverses

By restricting the domain of the tangent function to \(-\pi/2 < x < \pi/2\), we ensure the function is one-to-one, i.e., each output corresponds to exactly one input. This is necessary for the function to have an inverse, as multiple inputs corresponding to the same output would make the inverse function ambiguous.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Function Periodicity
Understanding the periodicity of the tangent function is crucial when studying trigonometry. The tangent function exhibits a repetitive pattern known as periodicity, with a period of \( \pi \), meaning that it repeats its behavior every \( \pi \) radians.

Here's a simple way to envision this: imagine the tangent function as a wave on the ocean. Every wave is similar and repeats at a regular interval鈥攖his interval for the tangent function is \( \pi \). However, for the inverse tangent function to exist, we must focus on just one wave. This helps in defining the function uniquely at every point within a specific interval.
Restricting Domain for Inverses
For a function to have an inverse, it must be one-to-one鈥攅very x-value should correspond to one unique y-value. The tangent function's all-encompassing wave pattern doesn't initially meet this condition because it repeats its values.

So, to craft an inverse, we restrict the domain; creating a single, non-repeating section of the wave. The interval \( (-\pi/2, \pi/2) \) is usually chosen for this purpose. Within this interval, the tangent function increases steadily from negative to positive infinity. Because the function doesn't loop back on itself here, we avoid any repetition, establishing the foundation needed for the inverse.
One-to-One Functions
A one-to-one function鈥攁lso called an injective function鈥攊s a type of function in which each x-value in the domain pairs with a unique y-value in the range. No y-value is associated with more than one x-value.

To put it more clearly, if you were assigned to give a different fruit to each of your classmates, ensuring no two classmates get the same fruit, you would be creating a one-to-one relationship. This concept is essential when we talk about inverse functions because only one-to-one functions can have inverses that are also functions.
Vertical Asymptotes
When exploring functions like the tangent, you will often encounter vertical asymptotes鈥攍ines that the function approaches but never touches. These are important in understanding why we restrict the domain of the tangent function for its inverse.

For the tangent function, vertical asymptotes occur at \( \pm\pi/2 \), and here鈥檚 how to picture them: imagine a fence that you can approach but never cross. These 'fences' are points where the tangent function goes off to infinity, creating a natural barrier for our domain restriction. By avoiding these asymptotes, we ensure the function remains well-behaved and set the stage for finding its inverse.

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Most popular questions from this chapter

In Exercises \(110-113,\) graph each pair of functions in the same viewing rectangle. Use your knowledge of the domain and range for the inverse trigonometric function to select an appropriate viewing rectangle. How is the graph of the second equation in each exercise related to the graph of the first equation? $$y=\cos ^{-1} x \text { and } y=\cos ^{-1}(x-1)$$

Solve and graph the solution set on a number line: $$x^{2}-4 x>-3$$

Exercises \(117-119\) will help you prepare for the material covered in the next section. In each exercise, complete the table of coordinates. Do not use a calculator. $$y=3 \sin \frac{\pi}{2} x$$ $$\begin{array}{|l|l|l|l|l|l|l|l|l|} \hline x & 0 & \frac{1}{3} & 1 & \frac{5}{3} & 2 & \frac{7}{3} & 3 & \frac{11}{3} & 4 \\ \hline y & & & & & & & \\ \hline \end{array}$$ After completing this table of coordinates, plot the nine ordered pairs as points in a rectangular coordinate system. Then connect the points with a smooth curve.

Carbon dioxide particles in our atmosphere trap heat and raise the planet's temperature. Even if all greenhousegas emissions miraculously ended today, the planet would continue to warm through the rest of the century because of the amount of carbon we have already added to the atmosphere. Carbon dioxide accounts for about half of global warming. The function $$y=2.5 \sin 2 \pi x+0.0216 x^{2}+0.654 x+316$$ models carbon dioxide concentration, \(y,\) in parts per million, where \(x=0\) represents January \(1960 ; x=\frac{1}{12},\) February \(1960 ; x=\frac{2}{12},\) March \(1960 ; \ldots, x=1,\) January \(1961 ; x=\frac{13}{12}\) February \(1961 ;\) and so on. Use a graphing utility to graph the function in a \([30,48,5]\) by \([310,420,5]\) viewing rectangle. Describe what the graph reveals about carbon dioxide concentration from 1990 through 2008 .

Exercises \(117-119\) will help you prepare for the material covered in the next section. In each exercise, complete the table of coordinates. Do not use a calculator. $$y=4 \sin \left(2 x-\frac{2 \pi}{3}\right)$$ $$\begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{X} & \frac{\pi}{3} & \frac{7 \pi}{12} & \frac{5 \pi}{6} & \frac{13 \pi}{12} & \frac{4 \pi}{3} \\ \hline \boldsymbol{y} & & & & & \\ \hline \end{array}$$
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