Question

Consider the following verbal description of a rational function. Where are the asymptotes and point of discontinuity located for the described rational function?

f of x equals the quotient of x plus four and x squared minus sixteen

(1 point)
Responses

vertical asymptote: y=0

horizontal asymptote: x=4

point of discontinuity: when x=−4

vertical asymptote: y is equal to 0 horizontal asymptote: x is equal to 4 point of discontinuity: when x is equal to negative 4

vertical asymptote: x=4

horizontal asymptote: y=0

point of discontinuity: when x=−4

vertical asymptote: x is equal to 4 horizontal asymptote: y is equal to 0 point of discontinuity: when x is equal to negative 4

vertical asymptote: x=4

horizontal asymptote: none

point of discontinuity: none

vertical asymptote: x is equal to 4 horizontal asymptote: none point of discontinuity: none

vertical asymptote: x=4

horizontal asymptote: y=12

point of discontinuity: none

Answer 1

To analyze the rational function given, \( f(x) = \frac{x + 4}{x^2 - 16} \), we first identify the features:

1. **Vertical Asymptotes**: These occur where the denominator is zero (but the numerator is not zero at those points). The denominator is \( x^2 - 16 \), which can be factored as \( (x - 4)(x + 4) \). Thus, the denominator is zero at \( x = 4 \) and \( x = -4 \).

2. **Point of Discontinuity**: A point of discontinuity occurs if the numerator is also zero at the same point as a zero of the denominator. Here, \( x = -4 \) makes the numerator \( x + 4 \) equal to zero as well, so there is a point of discontinuity at \( x = -4 \).

3. **Horizontal Asymptotes**: The horizontal asymptote of a rational function is determined by comparing the degrees of the numerator and denominator. Here, the numerator \( (x + 4) \) is degree 1, and the denominator \( (x^2 - 16) \) is degree 2. Since the degree of the denominator is greater than the degree of the numerator, there is a horizontal asymptote at \( y = 0 \).

Putting all of this together:

- Vertical asymptote: \( x = 4 \)
- Horizontal asymptote: \( y = 0 \)
- Point of discontinuity: \( x = -4 \)

The correct response matches these findings:

"vertical asymptote: x = 4; horizontal asymptote: y = 0; point of discontinuity: when x = -4."