Given \( f(x)=\sqrt{x+9} \) and \( g(x)=27 x-18 \) find \( (f \circ g)(x) \). \( \begin{array}{l}3 \sqrt{3 x-1} \\ 9 \sqrt{3 x-1} \\ 3 \sqrt{3 x+1} \\ 9 \sqrt{3 x}\end{array} \)

To find \( (f \circ g)(x) \), we need to evaluate \( f(g(x)) \). First, calculate \( g(x) \): \[ g(x) = 27x - 18 \] Next, substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(27x - 18) = \sqrt{(27x - 18) + 9} = \sqrt{27x - 9} \] Now, simplify \( \sqrt{27x - 9} \): \[ \sqrt{27x - 9} = \sqrt{9(3x - 1)} = 3\sqrt{3x - 1} \] So, \( (f \circ g)(x) = 3\sqrt{3x - 1} \). The correct answer is \( \boxed{3 \sqrt{3 x - 1}} \). The function \( g(x) = 27x - 18 \) is a linear transformation, shifting input values in a linear manner, while \( f(x) = \sqrt{x + 9} \) applies a square root operation, changing the function's growth rate from linear to square-root growth. This interplay showcases how a linear function can dramatically change the nature of a subsequent non-linear function. In real-world contexts, functions composed like this can represent various processes such as financial growth or physical phenomena, where an initial linear change (like investment over time) later encounters diminishing returns as governed by a square root function, illustrating that not all growth is exponential or linear indefinitely!