Describe how to find the domain and range for the tangent function.

To find the domain and range of the tangent function, we can follow these steps: ### Step 1: Understanding the Tangent Function The tangent function is defined as: \[ \tan(x) = \frac{\sin(x)}{\cos(x)} \] This means that the tangent function is undefined wherever the cosine function is equal to zero, as division by zero is not allowed. ### Step 2: Finding the Domain 1. **Identify where \(\cos(x) = 0\)**: The cosine function is equal to zero at: \[ x = \frac{\pi}{2} + k\pi \quad \text{for } k \in \mathbb{Z} \] This means that the tangent function is undefined at these points. 2. **Express the Domain**: The domain of the tangent function can be expressed as: \[ \text{Domain: } x \in \mathbb{R} \setminus \left\{ \frac{\pi}{2} + k\pi \mid k \in \mathbb{Z} \right\} \] In simpler terms, the domain includes all real numbers except for the points where \(x\) is an odd multiple of \(\frac{\pi}{2}\). ### Step 3: Finding the Range 1. **Behavior of the Tangent Function**: The tangent function can take on any real value. As \(x\) approaches the points where the function is undefined (i.e., \(\frac{\pi}{2} + k\pi\)), the value of \(\tan(x)\) approaches \(\infty\) or \(-\infty\). 2. **Express the Range**: Therefore, the range of the tangent function is: \[ \text{Range: } y \in \mathbb{R} \] This means that the tangent function can output any real number. ### Summary - **Domain**: \(x \in \mathbb{R} \setminus \left\{ \frac{\pi}{2} + k\pi \mid k \in \mathbb{Z} \right\}\) - **Range**: \(y \in \mathbb{R}\) This concludes the process of finding the domain and range of the tangent function.