To analyze the function \( f(x) = \frac{9x^2 - 36}{3x + 6} \), we begin by simplifying the expression.
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Factoring the numerator: The numerator, \( 9x^2 - 36 \), is a difference of squares, which can be factored as follows: \[ 9x^2 - 36 = 9(x^2 - 4) = 9(x - 2)(x + 2) \]
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Factoring the denominator: The denominator, \( 3x + 6 \), can be factored as: \[ 3x + 6 = 3(x + 2) \]
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Simplifying the function: Thus, we can rewrite the function: \[ f(x) = \frac{9(x - 2)(x + 2)}{3(x + 2)} = 3(x - 2) \] for \( x \neq -2 \), because at \( x = -2 \) the original function is undefined due to division by zero.
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Identifying the discontinuity: Since there is a factor \( x + 2 \) in both the numerator and denominator, the function is simplified to \( f(x) = 3(x - 2) \) for all \( x \) except \( x = -2 \). Therefore, there is a discontinuity at \( x = -2 \).
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Evaluating the value at the discontinuity: At the discontinuity \( x = -2 \), we can evaluate: \[ f(-2) = 3(-2 - 2) = 3(-4) = -12 \]
Now that we have simplified the function and determined that there is a discontinuity at \( x = -2 \) with a corresponding value of \( y = -12 \), we can identify the correct graph.
The correct choice based on this analysis is:
- Graph of \( 3x - 6 \), with discontinuity at \( -2, -12 \).