Let's analyze the function [tex]\( f(x) = (x - 3)(x + 1) \)[/tex].
1. Identify the Roots:
The function touches the x-axis where it equals zero. Solving for [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
(x - 3)(x + 1) = 0
\][/tex]
gives us the roots [tex]\( x = -1 \)[/tex] and [tex]\( x = 3 \)[/tex].
2. Analyze the Parabola:
The function is a quadratic equation and opens upwards because the coefficient of [tex]\( x^2 \)[/tex] in the expanded form (which would be [tex]\( x^2 - 2x - 3 \)[/tex]) is positive.
3. Determine Intervals of Positivity and Negativity:
The intervals change sign at the roots. We analyze the sign of the function in the intervals:
- For [tex]\( x < -1 \)[/tex]: Choose a test point like [tex]\( x = -2 \)[/tex].
[tex]\[
f(-2) = (-2 - 3)(-2 + 1) = (-5)(-1) = 5 \quad \text{(positive)}
\][/tex]
- For [tex]\( -1 < x < 3 \)[/tex]: Choose a test point like [tex]\( x = 0 \)[/tex].
[tex]\[
f(0) = (0 - 3)(0 + 1) = (-3)(1) = -3 \quad \text{(negative)}
\][/tex]
- For [tex]\( x > 3 \)[/tex]: Choose a test point like [tex]\( x = 4 \)[/tex].
[tex]\[
f(4) = (4 - 3)(4 + 1) = (1)(5) = 5 \quad \text{(positive)}
\][/tex]
So the function is positive for [tex]\( x < -1 \)[/tex] and [tex]\( x > 3 \)[/tex], and negative for [tex]\( -1 < x < 3 \)[/tex].
4. Determine Intervals of Decrease:
Finding the vertex of the parabola helps us understand where the function increases or decreases. The vertex lies midway between the roots:
[tex]\[
x = \frac{-1 + 3}{2} = 1
\][/tex]
The function decreases from [tex]\( -\infty \)[/tex] to the vertex at [tex]\( x = 1 \)[/tex] and increases from [tex]\( x = 1 \)[/tex] to [tex]\( + \infty \)[/tex].
5. Intersection of Positivity and Decrease:
The function is positive and decreasing where the intervals of positivity and decrease intersect:
- The function is positive in [tex]\( x < -1 \)[/tex] and [tex]\( x > 3 \)[/tex].
- The function is decreasing in [tex]\( x < 1 \)[/tex].
The intersection of these intervals is [tex]\( x < -1 \)[/tex].
So, the values for which the graph is both positive and decreasing are:
[tex]\[
\boxed{\text{all real values of } x \text{ where } x < -1}
\][/tex]
