Domain Analysis: Finding Where (x-1/3)(-x-2¼) > 0

Question

Find the positive and negative domains of the function:

y=(x13)(x214) y=\left(x-\frac{1}{3}\right)\left(-x-2\frac{1}{4}\right)

Determine for which values of x x the following is true:

f(x) > 0

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the roots of the quadratic expression.
  • Step 2: Determine the sign of each factor in intervals defined by these roots.
  • Step 3: Identify where the product of these factors is positive.

Now, let's work through each step:

Step 1: Identify the roots.
The given function is y=(x13)(x214) y = \left(x - \frac{1}{3}\right)\left(-x - 2\frac{1}{4}\right) . To find the roots, solve each factor for zero:

  • x13=0 x - \frac{1}{3} = 0 gives x=13 x = \frac{1}{3} .
  • x214=0-x - 2\frac{1}{4} = 0 gives x=214 x = -2\frac{1}{4} .

Step 2: Determine the sign of each factor in the intervals defined by these roots.
The zeros divide the x-axis into three intervals: (,214)(-∞, -2\frac{1}{4}), (214,13)(-2\frac{1}{4}, \frac{1}{3}), and (13,)(\frac{1}{3}, ∞).

Step 3: Test the signs and find where the product is positive.

  • For x<214 x < -2\frac{1}{4} , test with x=3 x = -3 : - x13=313<0 x - \frac{1}{3} = -3 - \frac{1}{3} < 0 - x214=3214>0-x - 2\frac{1}{4} = 3 - 2\frac{1}{4} > 0 - Product: Negative
  • For 214<x<13 -2\frac{1}{4} < x < \frac{1}{3} , test with x=0 x = 0 : - x13=013<0 x - \frac{1}{3} = 0 - \frac{1}{3} < 0 - x214=0214>0-x - 2\frac{1}{4} = 0 - 2\frac{1}{4} > 0 - Product: Positive
  • For x>13 x > \frac{1}{3} , test with x=1 x = 1 : - x13=113>0 x - \frac{1}{3} = 1 - \frac{1}{3} > 0 - x214=1214<0-x - 2\frac{1}{4} = -1 - 2\frac{1}{4} < 0 - Product: Negative

Therefore, the solution to f(x)>0 f(x) > 0 is in the interval:

214<x<13 -2\frac{1}{4} < x < \frac{1}{3} .

Answer

-2\frac{1}{4} < x < \frac{1}{3}