Parabola Domain Analysis: Finding x-Values Where f(x) > 0

The graph of the function below does not intersect the $x$-axis.

The parabola's vertex is marked A.

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

To solve this problem, let's analyze the key characteristics of the parabola:

• Since the parabola does not intersect the $x$-axis, it indicates that it is entirely either above or below the $x$-axis.
• The graph of a parabola $ax^2 + bx + c$ does not intersect the $x$-axis when its discriminant $b^2 - 4ac$ is negative. Thus, it does not have any real roots.
• If the parabola opens upwards, then the function is entirely above the $x$-axis if $a > 0$ and below if $a < 0$.
• Given the problem indicates the parabola never reaches or crosses the $x$-axis and the absence of real roots, a positive opening parabola cannot reach positive territory in when not intersecting the x-axis.

Since the parabola's graph neither touches nor crosses the $x$-axis and isn't stated to be always positive or negative, we conclude:

The function does not have a positive domain.