Math Problem Statement

Указать функцию, график которой представлен на рисунке.

Solution

The graph in the image shows a function that is periodic and includes a discontinuity at x=0x = 0. Based on its shape, we can identify the function as:

f(x)=tan(x)f(x) = \tan(x)

Key features of the graph:

  1. The function is undefined at x=±π2,±3π2,x = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \dots, corresponding to the vertical asymptotes.
  2. The graph exhibits periodicity with a period of π\pi.
  3. At x=0x = 0, the function value is 0.
  4. The behavior of the function aligns with the tangent's steep increase and vertical asymptotes.

Would you like further details about the function or its properties? Here are some related questions:

  1. What is the domain and range of the tangent function?
  2. How can the tangent function be derived from sine and cosine?
  3. What are the asymptotes of the tangent function, and how are they calculated?
  4. How does the period of tan(x)\tan(x) compare with other trigonometric functions?
  5. How would the graph of tan(x)\tan(x) change if a transformation (e.g., vertical stretch or shift) is applied?

Tip: Remember that the tangent function is not defined at points where cos(x)=0\cos(x) = 0, as tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}.

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Math Problem Analysis

Mathematical Concepts

Trigonometric Functions
Graph Analysis
Periodicity

Formulas

f(x) = tan(x)
tan(x) = sin(x) / cos(x)

Theorems

Periodicity of Trigonometric Functions
Asymptotes of Tangent Function

Suitable Grade Level

Grades 10-12