Math Problem Statement

Determine the vertical asymptotes of the function f from the given graph. If there are multiple, enter your answer as a comma-separated list.

Solution

The graph provided shows a rational function with vertical asymptotes, which are the vertical lines where the function tends toward infinity or negative infinity. Vertical asymptotes occur when the denominator of the rational function equals zero and the numerator is non-zero.

From the graph, the function seems to have vertical asymptotes at the following values of xx:

  • x=2x = -2
  • x=1x = 1

These are the points where the function's graph shoots up or down without crossing those lines.

The vertical asymptotes of the function are:

x=2,1x = -2, 1

Would you like further details on how vertical asymptotes are determined for rational functions, or any other related concepts?


Here are 5 questions to expand your understanding:

  1. How do you determine vertical asymptotes from the equation of a rational function?
  2. What happens to the function's value as it approaches a vertical asymptote?
  3. Can rational functions have both vertical and horizontal asymptotes?
  4. How are vertical asymptotes related to the denominator of a rational function?
  5. How do you distinguish vertical asymptotes from holes in a graph?

Tip: To find vertical asymptotes in rational functions, focus on the values of xx that make the denominator zero but not the numerator.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Asymptotes

Formulas

Vertical Asymptotes: Values of x where the denominator equals zero and the numerator is non-zero.

Theorems

Concept of Asymptotes
Properties of Rational Functions

Suitable Grade Level

Grades 10-12