Find functions \( f \) and \( g \) so that \( f \circ g=H \). H(x) \( f 6 x+6 \mid \) 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 the correct pair of functions \( f \) and \( g \) such that \( f \circ g = H(x) = 6x + 6 \), we need to first express what \( f(g(x)) \) should equal. Let's analyze each option: A. \( f(x)=\frac{x-6}{6}, g(x)=|-x| \) - Here, \( g(x) = |-x| \) makes \( g(x) \) always non-negative. Plugging this into \( f \), we get: \[ f(g(x)) = f(|-x|) = \frac{|-x| - 6}{6} \] This does not simplify to \( 6x + 6 \), so A is incorrect. B. \( f(x)=|x|, g(x)=6x+6 \) - Plugging in \( g(x) \): \[ f(g(x)) = f(6x + 6) = |6x + 6| \] This cannot simplify to \( 6x + 6 \) for negative \( x \), so B is incorrect. C. \( f(x)=|-x|, g(x)=\frac{x-6}{6} \) - Plugging in \( g(x) \): \[ f(g(x)) = f\left(\frac{x-6}{6}\right) = \left|-\frac{x-6}{6}\right| = \frac{x-6}{6} \] This does not equal \( 6x + 6 \), so C is incorrect. D. \( f(x)=6x+6, g(x)=|x| \) - Plugging in \( g(x) \): \[ f(g(x)) = f(|x|) = 6|x| + 6 \] Again, this does not simplify directly to \( 6x + 6 \) since it depends on the sign of \( x \). After going through each option, it appears none of the pairs correctly result in \( f \circ g = H(x) = 6x + 6 \). However, Option D seems to be a plausible candidate if we are only looking for non-negative \( x \) inputs. So, while strictly none of the options works for all \( x \), D would be the most reasonable choice for positive inputs, effectively outputting \( H(x) \) when \( x \ge 0 \).