Math Problem Statement

Find the vertical asymptotes. Select the correct choice below and fill in any answer boxes within your choice.

A.

The vertical​ asymptote(s) can be described by the​ line(s)

enter your response here.

​(Type an equation. Use a comma to separate answers as​ needed.)

B.

There are no vertical asymptotes.

Solution

To find the vertical asymptotes of the function f(x)=2xx+4f(x) = \frac{2x}{x + 4}, follow these steps:

Step 1: Identify potential vertical asymptotes.

  • Vertical asymptotes occur where the denominator of a rational function is equal to zero (and the numerator is non-zero at that point).

Step 2: Set the denominator equal to zero.

  • The denominator of f(x)=2xx+4f(x) = \frac{2x}{x + 4} is x+4x + 4.

Set the denominator equal to zero and solve for xx: x+4=0x + 4 = 0 x=4x = -4

Conclusion:

  • There is a vertical asymptote at x=4x = -4.

Choice A is the correct answer:

  • The vertical asymptote(s) can be described by the line x=4x = -4.

Would you like further details on this, or do you have any other questions? Here are five related questions:

  1. What happens to f(x)f(x) as xx approaches 4-4 from the left and right?
  2. Can a function have more than one vertical asymptote?
  3. How does the presence of vertical asymptotes affect the graph of a function?
  4. What is the difference between a vertical asymptote and a hole in the graph?
  5. How do you determine if a vertical asymptote exists using limits?

Tip: Vertical asymptotes represent values of xx where the function becomes undefined and the function value tends to ±\pm \infty.

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Math Problem Analysis

Mathematical Concepts

Functions
Rational Functions
Vertical Asymptotes

Formulas

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Theorems

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Suitable Grade Level

High School