Find functions \( f \) and \( g \) so that \( f \circ g=H \) \( H(x)=|6 x+6| \) Choose the correct pair of functions A. \( f(x)=\frac{x-6}{6}, g(x)=|-x| \) B. \( f(x)=|x|, g(x)=6 x+6 \) C. \( f(x)=|-x|, g(x)=\frac{x-6}{6} \) D. \( f(x)=6 x+6, g(x)=|x| \)

To find functions \( f \) and \( g \) such that \( f \circ g = H \) where \( H(x) = |6x + 6| \), we need to analyze the composition of functions in the given options. Let's look at option B: - Choose \( g(x) = 6x + 6 \). - Then, \( f(x) = |x| \). Now, we compute \( f(g(x)) \): - \( g(x) = 6x + 6 \). - So, \( f(g(x)) = f(6x + 6) = |6x + 6| \). This matches \( H(x) \), confirming that option B is correct. Let's check the other options: - Option A: \( f(x) = \frac{x-6}{6}, g(x) = |-x| \) gives \( f(g(x)) \) as \( \frac{|-x|-6}{6} \), which does not equal \( |6x+6| \). - Option C: \( f(x) = |-x|, g(x) = \frac{x-6}{6} \) does not satisfy \( |6x+6| \). - Option D: \( f(x) = 6x + 6, g(x) = |x| \) gives \( f(g(x)) = 6|x| + 6 \neq |6x + 6| \). Thus, the only correct pair of functions is: **B. \( f(x) = |x|, g(x) = 6x + 6 \)**.