Math Problem Statement

Let r(x) = 1 / (x + 8). This function has a vertical asymptote at x = -8. What is the equation of the vertical asymptote of r(-8x)?

Solution

The given function is:

r(x)=1x+8r(x) = \frac{1}{x + 8}

The vertical asymptote of a rational function occurs when the denominator is equal to zero, and the function becomes undefined.

For r(x)=1x+8r(x) = \frac{1}{x + 8}, the denominator x+8=0x + 8 = 0 gives:

x=8x = -8

So, the original function has a vertical asymptote at x=8x = -8.

The question is about finding the vertical asymptote of the function r(8x)r(-8x).

Let's rewrite the function r(8x)r(-8x):

r(8x)=18x+8r(-8x) = \frac{1}{-8x + 8}

To find the vertical asymptote, set the denominator to zero:

8x+8=0-8x + 8 = 0

Solving for xx:

8x=8-8x = -8

x=1x = 1

Thus, the equation of the vertical asymptote of r(8x)r(-8x) is:

x=1x = 1

Would you like more details on the steps involved or have any questions?

Here are some related questions you might consider:

  1. What is a vertical asymptote, and why does it occur in rational functions?
  2. How does the transformation f(x)f(-x) affect the graph of a function?
  3. How do you find horizontal asymptotes of rational functions?
  4. What happens to the asymptotes if you multiply or divide the function by a constant?
  5. How do you determine the behavior of a function near its asymptotes?

Tip: To find vertical asymptotes, always look for the values of xx that make the denominator zero, since the function is undefined there.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Vertical Asymptotes
Function Transformations

Formulas

The vertical asymptote occurs when the denominator of a rational function equals zero

Theorems

Vertical Asymptote Theorem
Function Transformation (r(ax))

Suitable Grade Level

Grades 10-12