To analyze the function \( f(x) = \frac{x + 4}{x^2 - 16} \), we need to determine the asymptotes and points of discontinuity.
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Factor the denominator: \[ x^2 - 16 = (x - 4)(x + 4) \] This means that the function can be simplified as follows: \[ f(x) = \frac{x + 4}{(x - 4)(x + 4)} \]
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Identify the point of discontinuity: The function has a factor of \( x + 4 \) in both the numerator and denominator. This indicates a removable discontinuity at \( x = -4 \). Therefore, there is a point of discontinuity at \( x = -4 \).
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Identify vertical asymptotes: Vertical asymptotes occur where the denominator is zero but the numerator is not zero. Thus:
- The denominator \( (x - 4)(x + 4) = 0 \) at \( x = 4 \) and \( x = -4 \).
- Since \( x = -4 \) is a removable discontinuity (as discussed), the vertical asymptote will only occur at \( x = 4 \).
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Identify horizontal asymptotes: To find horizontal asymptotes, we look at the degrees of the numerator and denominator. In this case, the degree of the numerator is 1 and the degree of the denominator is 2. For rational functions:
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
Now, summarizing this information:
- Vertical asymptote: \( x = 4 \)
- Horizontal asymptote: \( y = 0 \)
- Point of discontinuity: \( x = -4 \)
Given this analysis, the correct response is:
vertical asymptote: \( x = 4 \); horizontal asymptote: \( y = 0 \); point of discontinuity: when \( x = -4 \).
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