Question

analyze the graph of the function f(x) to complete the statement. f(x)<0 over (-∞, -3) and what other interval? (-2.4, -1.1) (-3, -1.1) (-1.1, 2) (-1.1, 0.9)

Explanation:

Step1: Understand the function's sign

The function \( f(x)<0 \) when the graph is below the \( x - \)axis.

Step2: Analyze each interval

• For the interval \((-2.4,-1.1)\): The graph is above the \( x - \)axis (since \( f(-2.4)=17>0 \)), so \( f(x)>0 \) here.
• For the interval \((-3,-1.1)\): The graph is above the \( x - \)axis (at \( x=-3,f(-3) = 0\) and moving from \( x=-3 \) to \( x=-1.1 \), the part of the graph in this sub - interval (excluding \( x = - 3\)) is above the \( x - \)axis).
• For the interval \((-1.1,2)\): The graph is above the \( x - \)axis at \( x = 2\) (not in the domain we are considering for \( f(x)<0 \) based on roots).
• For the interval \((-1.1,0.9)\): The roots of the function are \( x=-3,x=-1.1,x = 0.9\). The function \( y = f(x) \) is a polynomial (assumed from the graph's shape). A polynomial changes its sign at its roots. Since \( f(x)<0 \) for \( x\in(-\infty,-3)\) (left - most root) and between \( x=-1.1\) and \( x = 0.9\) (because the graph is below the \( x - \)axis in this interval as we can see from the graph's position relative to the \( x - \)axis).

Answer:

\((-1.1,0.9)\)