Problem 3 - Algebra 2 and Trigonometry

Question

Sketch the graph of \(y=\tan x\) from \(x=-\frac{3 \pi}{2}\) to \(x=\frac{3 \pi}{2}\) a. What is the period of \(y=\tan x ?\) b. What is the domain of \(y=\tan x ?\) c. What is the range of \(y=\tan x ?\)

Step-by-Step Solution

Answer

a. The period is \( \pi \). b. Domain: \( x \in \mathbb{R} \setminus \{ -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2} \} \). c. Range: \( \mathbb{R} \).

Step 1: Identify the Period of the Function

The period of the tangent function, \( y = \tan{x} \), is \( \pi \). This means that the function repeats every \( \pi \) units along the x-axis. For this specific function, one complete cycle occurs from \( x = 0 \) to \( x = \pi \) or from \( x = -\frac{\pi}{2} \) to \( x = \frac{\pi}{2} \).

Step 2: Determine the Domain

The function \( y = \tan{x} \) is undefined at points where \( \cos{x} = 0 \) because \( \tan{x} = \frac{\sin{x}}{\cos{x}} \). These points occur at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. For the interval \( x = -\frac{3\pi}{2} \) to \( x = \frac{3\pi}{2} \), the function is undefined at \( x = -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \text{ and } \frac{3\pi}{2} \). Thus, the domain is \( x \in \mathbb{R} \setminus \{ -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2} \} \).

Step 3: Determine the Range

The range of \( y = \tan{x} \) is all real numbers (\( \mathbb{R} \)). This is because as \( x \) approaches the points of discontinuity (vertical asymptotes), \( \tan{x} \) tends toward positive or negative infinity.

Key Concepts

Tangent Function

The tangent function, denoted as \( y = \tan x \), is a fundamental trigonometric function that is defined as the ratio of the sine and cosine functions: \( \tan x = \frac{\sin x}{\cos x} \).
This function takes a given angle and provides a value based on this ratio. As the function's input (angle \( x \)) approaches certain values, the denominator (cosine) approaches zero, causing the tangent value to increase or decrease without bound. These points of discontinuity where \( \cos x = 0 \) are called vertical asymptotes, leading to the tangent function being undefined at these points.
The tangent function is periodic and is well-known for having vertical asymptotes, creating a characteristic wave-like graph that extends infinitely.

Period of a Function

The period of a function refers to the interval at which the function repeats its pattern. For the tangent function, \( y = \tan x \), the period is \( \pi \).
This implies that every \( \pi \) units, the tangent function exhibits the same behavior, repeating its wave-like pattern and vertical asymptotes.
Identifying the period of a function is crucial for understanding and predicting its graph. For example, if you know one period of the tangent, you can predict what the graph looks like by repeating this pattern across the x-axis. In this specific exercise, the given range from \( x = -\frac{3\pi}{2} \) to \( x = \frac{3\pi}{2} \) includes three periods of the tangent function.

Domain and Range

When discussing functions, the domain refers to all possible input values (x-values), whereas the range refers to all possible output values (y-values).

• **Domain**: For the tangent function, \( y = \tan x \), any point where \( \cos x = 0 \) must be excluded because these points make the function undefined. This results in vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \). For the interval from \( x = -\frac{3\pi}{2} \) to \( x = \frac{3\pi}{2} \), the points of discontinuity are \( -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \text{ and } \frac{3\pi}{2} \).


• **Range**: The range of the tangent function is all real numbers, \( \mathbb{R} \). Unlike some other trigonometric functions, the output for \( \tan x \) is not bounded and can span from negative to positive infinity. This stems from the function's behavior near its vertical asymptotes, where it tends towards positive or negative infinity.

Understanding the domain and range helps in sketching accurate graphs and in solving equations involving the tangent function. This enables students to grasp the boundaries and possible outputs of the function easily.