Given the function:
Determine for which values of x the following is true:
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Given the function:
Determine for which values of x the following is true:
To determine for which values of the function is positive, we solve the inequality:
We start by finding the roots of the associated quadratic equation:
This can be rewritten as:
Solving for , we have:
The roots are and . These points are where the parabola intersects the x-axis. Since the parabola opens downwards (as the coefficient of is negative), the function is positive between the roots.
Therefore, the function over the interval:
Thus, the correct choice is:
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 \( f\left(x\right) > 0 \).
Because the coefficient of is negative (-1), this parabola opens downward. The function is positive where the parabola is above the x-axis, which is between the roots.
Look at the coefficient of : if it's positive, the parabola opens upward (smile). If it's negative, it opens downward (frown).
Always test a point! Pick any number in your suspected interval (like x = 0) and substitute it into the original function. If the result matches your inequality, you're correct.
No! The inequality is (strictly greater than), so we use open intervals. At x = ±7, the function equals zero, not greater than zero.
Absolutely! Graph and look for where the parabola is above the x-axis. The x-values of that region give you your answer.
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