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

Find the positive and negative domains of the function below:

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

Then determine for which values of x x the following is true:

f(x)<0 f(x) < 0

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Find the positive and negative domains of the function below:

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

Then determine for which values of x x the following is true:

f(x)<0 f(x) < 0

2

Step-by-step solution

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: x13=0 x - \frac{1}{3} = 0 gives x=13 x = \frac{1}{3} .
    Solve x214=0 -x - 2\frac{1}{4} = 0 gives x=214 x = -2\frac{1}{4} or x=94 x = -\frac{9}{4} .
  • Step 3: This defines the critical points, x=13 x = \frac{1}{3} and x=94 x = -\frac{9}{4} .
  • Step 4: Determine the sign of the function on intervals: (,94) (-\infty, -\frac{9}{4}) , (94,13) (-\frac{9}{4}, \frac{1}{3}) , and (13,) (\frac{1}{3}, \infty) .
  • Step 5: Test points in each interval:
    For x<94 x < -\frac{9}{4} , both factors are negative, the product is positive: Interval does not satisfy.
    For 94<x<13 -\frac{9}{4} < x < \frac{1}{3} , signs will vary, and the product is negative: Interval satisfies f(x)<0 f(x) < 0 .
    For x>13 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: 214<x<13 -2\frac{1}{4} < x < \frac{1}{3} .

The correct answer choice is therefore Choice 1

3

Final Answer

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

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of \( x \) where \( f\left(x\right) > 0 \).

AAABBBCCCX

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