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 first need to define the function \( (f+g)(x) \). Given: - \( f(x) = |x| + 9 \) - \( g(x) = -6 \) We can express \( (f+g)(x) \) as follows: \[ (f+g)(x) = f(x) + g(x) = (|x| + 9) + (-6) = |x| + 3 \] Next, we need to determine the range of \( |x| + 3 \). 1. The absolute value function \( |x| \) is always non-negative, meaning \( |x| \geq 0 \) for all \( x \). 2. Therefore, the minimum value of \( |x| + 3 \) occurs when \( |x| = 0 \): \[ |x| + 3 \geq 0 + 3 = 3 \] 3. As \( |x| \) increases, \( |x| + 3 \) will also increase without bound. Thus, the maximum value is not limited. From this analysis, we conclude that: \[ |x| + 3 \geq 3 \text{ for all values of } x \] Thus, the range of \( (f+g)(x) \) is \( [3, \infty) \). The correct description of the range of \( (f+g)(x) \) is: \[ (f+g)(x) \geq 3 \text{ for all values of } x \]