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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 function is typically \(-蟺/2 < x < 蟺/2\).

Step by step solution

01

Understanding Inverse Functions

An inverse function is a function that undoes the work of another function. In other words, if you start with an input, apply a function, and then apply its inverse function, you return back to the initial input. Not all functions have inverse functions. For a function to have an inverse, it must be bijective, meaning that it is both injective(one-to-one) and surjective(onto). This implies that each output value has exactly one associated input value.
02

Understanding the Tangent Function

The tangent function is commonly defined as \(\frac{sin(x)}{cos(x)}\). From this definition, it can be seen that \(\tan(x)\) has vertical asymptotes at \(x = \pi/2 + n\pi\) as \(\cos(x)\) equals zero at these values and the value of \(\tan(x)\) tends to infinity. As a result, \(\tan(x)\) isn't bijective over its entire domain and doesn't have an inverse function in its current form.
03

Restricting the Tangent Function

To restrict the tangent function to a domain where it is bijective, it鈥檚 necessary to choose a specific interval of x-values so that each y-value of the function is associated with exactly one x-value. One common choice of restriction for \(\tan(x)\) to make it bijective is the open interval \((-蟺/2, 蟺/2)\). In this interval, the function \(\tan(x)\) is continuous and strictly increasing, meeting the criteria for it to have an inverse.

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