To analyze the function \( f(x) = (x - 4)(x + 1) \), we can first find its critical points and determine the intervals where it is increasing or decreasing.
Step 1: Find the Roots
The roots (x-intercepts) of the function can be found by setting the function to zero: \[ f(x) = (x - 4)(x + 1) = 0 \] This gives us the roots \( x = -1 \) and \( x = 4 \).
Step 2: Determine the Vertex
The vertex of a parabola given by a quadratic function in standard form, \( ax^2 + bx + c \), can be calculated using: \[ x_v = -\frac{b}{2a} \] To convert \( f(x) \) into standard form, we expand: \[ f(x) = x^2 - 3x - 4 \] Here, \( a = 1 \) and \( b = -3 \). Thus: \[ x_v = -\frac{-3}{2 \times 1} = \frac{3}{2} = 1.5 \] Now that we know the x-coordinate of the vertex is at \( x = 1.5 \), we can compute the function value at the vertex to find its maximum or minimum point.
Step 3: Determine the Function's Behavior
The parabola opens upwards (since \( a > 0 \)), indicating that the vertex is a minimum point.
The function will be:
- Decreasing on the interval \( (-\infty, x_v) \) or \( (-\infty, 1.5) \).
- Increasing on the interval \( (x_v, \infty) \) or \( (1.5, \infty) \).
Step 4: Evaluate the Statements
- The function is increasing for all real values of x where \( x < 0 \): False (it's decreasing for \( x < 1.5 \)).
- The function is increasing for all real values of x where \( x < -1 \) and where \( x > 4 \): False (it’s decreasing to the left of \( x = 1.5 \)).
- The function is decreasing for all real values of x where \( -1 < x < 4 \): True (within this interval \( x = 4 \) onwards, it starts increasing).
- The function is decreasing for all real values of x where \( x < 1.5 \): True but not the most precise since it is only decreasing until reaching the vertex, after which it begins increasing.
Conclusion
The correct statement is: The function is decreasing for all real values of x where \( -1 < x < 4 \).
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