To analyze where the function \( f(x) = (x - 3)(x + 1) \) is positive and decreasing, let's break it down:
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Identify the roots: The function has roots at \( x = -1 \) and \( x = 3 \) because that is where \( f(x) = 0 \).
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Vertex of the parabola: The vertex of the parabola \( f(x) = (x - 3)(x + 1) \) occurs at \( x = 1 \), where \( f(1) = (1 - 3)(1 + 1) = (-2)(2) = -4 \). The vertex is at the point \( (1, -4) \).
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Nature of the parabola: Since the parabola opens upwards (the coefficient of \( x^2 \) is positive), it will be:
- Negative between the roots \( (-1, 3) \)
- Positive outside the roots (i.e., when \( x < -1 \) and \( x > 3 \))
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Determine where the graph is decreasing:
- The function decreases from \( x = -1 \) to \( x = 1 \) (it reaches its minimum at the vertex).
- The function starts increasing again after the vertex at \( x = 1 \).
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Finding the interval:
- From the analysis:
- The graph is positive in the intervals \( x < -1 \) and \( x > 3 \).
- The graph is decreasing in the interval \( x < 1 \).
- The only overlap where the graph is both positive and decreasing is for \( x < -1 \).
- From the analysis:
Thus, the correct description for all of the values for which the graph is positive and decreasing is:
all real values of x where \( x < -1 \).