Problem 105 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The restriction on the tangent function required for it to have an inverse function is that its domain must be confined to an interval of width pi that does not include a vertical asymptote, such as the interval (-pi/2, pi/2). This makes the function one-to-one and thus invertible within this domain.

Step 1: Understand the graph of the tangent function

The tangent function, defined as tan(x) = sin(x) / cos(x), has a period of pi radians. This means that the graph of tan(x) repeats itself every pi radians. The function is undefined at odd multiples of pi/2, where the denominator cos(x) is zero, leading to vertical asymptotes.

Step 2: Visualize how to make the function one-to-one

To make tan(x) one-to-one and thus invertible, the domain must be restricted to an interval where the function is increasing or decreasing continuously without any vertical asymptotes. The most common choice for this interval is (-pi/2, pi/2), but any interval of breadth pi which does not include an asymptote would also work.

Step 3: Write down the restriction

The restriction on the tangent function so that it has an inverse function is thus that its domain must be confined to an interval of width pi that does not include a vertical asymptote. The most common choice for this interval is (-pi/2, pi/2). This means the tangent function is limited to producing unique outputs for inputs inside this range, making it one-to-one and invertible in this domain.