Find the positive and negative domains of the function:
y=(x−31)(−x−241)
Determine for which values of x the following is true:
f(x) > 0
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=(x−31)(−x−241). To find the roots, solve each factor for zero:
- x−31=0 gives x=31.
- −x−241=0 gives x=−241.
Step 2: Determine the sign of each factor in the intervals defined by these roots.
The zeros divide the x-axis into three intervals: (−∞,−241), (−241,31), and (31,∞).
Step 3: Test the signs and find where the product is positive.
- For x<−241, test with x=−3:
- x−31=−3−31<0
- −x−241=3−241>0
- Product: Negative
- For −241<x<31, test with x=0:
- x−31=0−31<0
- −x−241=0−241>0
- Product: Positive
- For x>31, test with x=1:
- x−31=1−31>0
- −x−241=−1−241<0
- Product: Negative
Therefore, the solution to f(x)>0 is in the interval:
−241<x<31.
-2\frac{1}{4} < x < \frac{1}{3}