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