Problem 65 - Precalculus Essentials

Question

Without drawing a graph, describe the behavior of the basic tangent curve.

Step-by-Step Solution

Answer

The basic tangent function has a domain of all real numbers excluding \(x = \frac{(2n+1)\pi}{2}\), a range of all real numbers, and a periodicity of \( \pi \). It is odd symmetric and its curve crosses the x-axis at \(x = n\pi\) and has vertical asymptotes at \(x = \frac{(2n+1)\pi}{2}\).

Step 1: Understanding the Domain of the Tangent function

To describe the behavior of a curve, start by defining its domain. For the tangent function, the set of all real numbers excluding the Values where the function is undefined can serve as the domain. The tangent function is undefined at values \(x = \frac{(2n+1)\pi}{2}\) where \(n\) is an integer. These correspond to points where the cosine function equals zero.

Step 2: Establishing the Range of the Tangent function

In this step, proceed to define the range of the function. Unlike the domain, the range of the tangent function is all real numbers. That is, for any value in the range, a corresponding value in the domain exists.

Step 3: Defining the Periodicity and Symmetry

The function tangent has a periodicity of \( \pi \), indicating that it repeats its values every \( \pi \) intervals. Additionally, it is odd symmetric - this means that for every point (x, y) on the tangent graph, the point (-x, -y) is also on the graph.

Step 4: Evaluate the Behavior at Special Points

Lastly, study the behavior of the tangent curve at its special points. At \(x = \frac{(2n+1)\pi}{2}\) (where the function is undefined), the function approaches positive or negative infinity. This indicates that the function has vertical asymptotes at these points. Also, the function crosses the x-axis at \(x = n\pi\), where n is an integer.