Question

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

Explanation:

Step1: Identify x-intercepts

The graph intersects the x - axis at \( x = 3 \) and \( x = 7 \).

Step2: Analyze where \( f(x)<0 \)

For a parabola opening downward (since the ends go down), the function is negative when \( x<3 \) or \( x > 7 \)? Wait, no. Wait, the graph: when \( x \) is less than 3, the graph is below the x - axis (negative), and when \( x \) is greater than 7, the graph is below the x - axis (negative). Wait, let's check the graph. The parabola crosses the x - axis at \( x = 3 \) and \( x=7 \). The vertex is above the x - axis. So the function \( f(x) \) is negative when \( x<3 \) or \( x > 7 \)? Wait, no, wait the left part: when \( x \) is less than 3, the graph is going down from the left? Wait, no, the graph: the two arms go down. So between \( x = 3 \) and \( x = 7 \), the graph is above the x - axis (positive), and outside that interval ( \( x<3 \) or \( x>7 \) ), the graph is below the x - axis (negative). Wait, let's see the y - values. When \( x = 0 \), the graph is below the x - axis (since at \( x = 0 \), y is negative). When \( x = 2 \), y is negative. When \( x = 8 \), y is negative. When \( x = 4 \), y is positive. So the domain where \( f(x) \) is negative is \( (-\infty,3)\cup(7,\infty) \)? Wait, but looking at the graph, the left side: the graph comes from below, crosses the x - axis at \( x = 3 \), goes up to the vertex, then comes down and crosses the x - axis at \( x = 7 \), then goes down. So yes, the function is negative when \( x<3 \) or \( x>7 \).

Wait, but maybe I made a mistake. Let's re - examine. The x - intercepts are at \( x = 3 \) and \( x = 7 \). The parabola opens downward (because the coefficient of \( x^{2} \) is negative, as the ends go down). For a downward - opening parabola, the function is positive between the two roots and negative outside the two roots. So \( f(x)>0 \) when \( 37 \).

Answer:

The domain on which \( f(x) \) is negative is \( (-\infty, 3)\cup(7, \infty) \) (or in interval notation, all real numbers less than 3 or greater than 7).