If \( f(x)=|x|+9 \) and \( g(x)=-6 \), which describes the range of \( (f+g)(x) \) ? \( \begin{array}{l}(f+g)(x) \geq 3 \text { for all values of } x \\ (f+g)(x) \leq 3 \text { for all values of } x \\ (f+g)(x) \leq 6 \text { for all values of } x \\ (f+g)(x) \geq 6 \text { for all values of } x\end{array} \)

To find the range of \( (f+g)(x) \), we start by calculating \( (f+g)(x) \). Given: \[ f(x) = |x| + 9 \] \[ g(x) = -6 \] Now, combining both functions: \[ (f+g)(x) = f(x) + g(x) = (|x| + 9) - 6 = |x| + 3 \] Next, we need to determine the range of \( |x| + 3 \). The absolute value function \( |x| \) is always non-negative, which means \( |x| \geq 0 \). Thus: \[ |x| + 3 \geq 3 \] This tells us that the minimum value of \( (f+g)(x) \) is 3. Since \( |x| \) can increase indefinitely, \( (f+g)(x) \) can take on any value greater than or equal to 3. Therefore, the correct description of the range of \( (f+g)(x) \) is: \[ (f+g)(x) \geq 3 \text{ for all values of } x \]