Parabola Analysis: Finding f(x) < 0 with Non-Intersecting X-Axis

The graph of the function below the 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 decide where $f(x) < 0$ for the given parabola, observe the following:

• The parabola does not intersect the x-axis, indicating it is either entirely above or below the x-axis.
• If the parabola were entirely above the x-axis for $f(x) > 0$, it would contradict the question by not giving a valid interval for $f(x) < 0$.
• Therefore, the correct conclusion is that the parabola is entirely below the x-axis, meaning $f(x) < 0$ for all $x$.

Based on the understanding of quadratic functions and their graph behavior, the function does not intersect the x-axis implies it is always negative.

Hence, the domain where $f(x) < 0$ is for all $x$. This leads us to choose:

The domain is always negative.