Question

determine the domain on which the following graph of $f(x)$ is positive.

Explanation:

Step1: Identify x-intercepts

The graph intersects the x - axis at \(x=-4\) and \(x = 0\).

Step2: Analyze the parabola's direction

The parabola opens upwards (since the coefficient of \(x^{2}\) is positive, as the arms of the parabola go up). For an upward - opening parabola, the function \(y = f(x)\) is positive when \(x\) is less than the smaller root or greater than the larger root.
The smaller root is \(x=-4\) and the larger root is \(x = 0\). So \(f(x)>0\) when \(x < - 4\) or \(x>0\).

Answer:

The domain on which \(f(x)\) is positive is \(x < - 4\) or \(x>0\) (in interval notation, \((-\infty,-4)\cup(0,\infty)\))