https://r2.erweima.ai/mathbot/image/172594217145003w2p4a4.jpg

funciones trigonometricas hacer ejemplo de  tan,---hacer el f(X), domf, ranf, periodo y la grafica

We are working with the function \( y = \tan(x) \), which is the tangent function. Based on the grid in the image, we are to complete the table, determine the domain, range, period, and plot the graph. Let’s go step by step.

### Step 1: Fill in the values of \( y = \tan(x) \)

The tangent function is defined as:

\[
y = \tan(x) = \frac{\sin(x)}{\cos(x)}
\]

The values for some key angles are as follows:

\[
\begin{array}{c|c}
x & \tan(x) \\
\hline
0 & 0 \\
\frac{\pi}{4} & 1 \\
\frac{\pi}{2} & \text{undefined} \quad (\text{vertical asymptote}) \\
\frac{3\pi}{4} & -1 \\
\pi & 0 \\
\frac{5\pi}{4} & 1 \\
\frac{3\pi}{2} & \text{undefined} \quad (\text{vertical asymptote}) \\
\frac{7\pi}{4} & -1 \\
2\pi & 0 \\
\end{array}
\]

Notice that at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), the tangent is undefined because the cosine function is zero at those points, which leads to a vertical asymptote in the graph.

### Step 2: Domain and Range of \( y = \tan(x) \)

- **Domain**: The tangent function is undefined where \( \cos(x) = 0 \), i.e., at \( x = \frac{\pi}{2} + n\pi \) where \( n \) is an integer. So, the domain is:
  \[
  \text{Domain}: x \in \mathbb{R} \quad \text{except} \quad x = \frac{\pi}{2} + n\pi, n \in \mathbb{Z}
  \]
  
- **Range**: The tangent function takes all real values as \( x \) approaches the vertical asymptotes. So, the range is:
  \[
  \text{Range}: (-\infty, \infty)
  \]

### Step 3: Period of the Tangent Function

The period of the tangent function is \( \pi \), which means that the function repeats itself every \( \pi \) units.

### Step 4: Graph of \( y = \tan(x) \)

To plot the graph of \( y = \tan(x) \), observe the following:
- At \( x = 0, \pi, 2\pi \), the value of \( y = 0 \) (these are the x-intercepts).
- At \( x = \frac{\pi}{4}, \frac{5\pi}{4} \), \( y = 1 \).
- At \( x = \frac{3\pi}{4}, \frac{7\pi}{4} \), \( y = -1 \).
- Vertical asymptotes occur at \( x = \frac{\pi}{2}, \frac{3\pi}{2} \).

Would you like me to plot the graph for you based on these values? Additionally, here are some related questions:

### Relative Questions:
1. How does the graph of \( y = \tan(x) \) differ from that of \( y = \sin(x) \) or \( y = \cos(x) \)?
2. What is the effect of transformations such as \( y = a \tan(bx + c) \) on the graph of \( y = \tan(x) \)?
3. How do the asymptotes of the tangent function change with transformations?
4. Can the tangent function ever be bounded within a specific range?
5. How does the period of other trigonometric functions compare with the period of the tangent function?

### Tip:
When plotting trigonometric functions, identifying key points like intercepts, maximum/minimum values, and asymptotes helps in constructing an accurate graph efficiently.

funciones trigonometricas hacer ejemplo de tan,---hacer el f(X), domf, ranf, periodo y la grafica

This example explores the tangent function y = tan(x), covering its domain, range, period, and key points. Learn how to graph y = tan(x), identifying intercepts, asymptotes, and the behavior of the function for various values of x.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation