Solve (x-1/3)(-x-2¼): Finding Negative Function Values and Domain

To solve this problem, we need to find the roots and determine the sign of the function on intervals between these roots:

• Step 1: Find the roots of the quadratic by solving each factor equal to zero.
• Step 2: $x - \frac{1}{3} = 0$ gives $x = \frac{1}{3}$.
  Solve $-x - 2\frac{1}{4} = 0$ gives $x = -2\frac{1}{4}$ or $x = -\frac{9}{4}$.
• Step 3: This defines the critical points, $x = \frac{1}{3}$ and $x = -\frac{9}{4}$.
• Step 4: Determine the sign of the function on intervals: $(-\infty, -\frac{9}{4})$, $(-\frac{9}{4}, \frac{1}{3})$, and $(\frac{1}{3}, \infty)$.
• Step 5: Test points in each interval:
  For $x < -\frac{9}{4}$, both factors are negative, the product is positive: Interval does not satisfy.
  For $-\frac{9}{4} < x < \frac{1}{3}$, signs will vary, and the product is negative: Interval satisfies $f(x) < 0$.
  For $x > \frac{1}{3}$, both factors are positive, the product is positive: Interval does not satisfy.

Thus, the solution is for values where the product is negative: $-2\frac{1}{4} < x < \frac{1}{3}$.

The correct answer choice is therefore Choice 1