Question

How to find holes and vertical asymptotes.

Answer 1

To find holes and vertical asymptotes in a function, factor the numerator and denominator, identify any common factors, determine potential vertical asymptotes by finding x-values that make the denominator zero, check if these potential asymptotes are canceled out by factors in the numerator, and determine if they become vertical asymptotes or holes.

Step-by-step explanation:

Finding Holes and Vertical Asymptotes

To find holes and vertical asymptotes in a function, follow these steps:

1. 
   
3. Factor the numerator and denominator of the function.
4. 
   
6. Identify any common factors that cancel out.
7. 
   
9. Determine the values of x that make the denominator equal to zero. These values are potential vertical asymptotes.
10. 
    
12. Check if any of these potential vertical asymptotes are canceled out by factors in the numerator.
13. 
    
15. If a potential vertical asymptote is not canceled out, it is a vertical asymptote. If it is canceled out, there is a hole at that x-value.

For example, let's consider the function f(x) = (x^2 - 4) / (x - 2).

- The numerator can be factored as (x - 2)(x + 2).
- The denominator is already factored as (x - 2).
- The factor (x - 2) cancels out in the numerator and denominator, leaving f(x) = x + 2.

Therefore, the function has a hole at x = 2, where the factor was canceled out.

There is no vertical asymptote in this example. Vertical asymptotes occur when the denominator factors cannot be canceled out by factors in the numerator.