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. $$ f(x)=6 x+6, g(x)=|x| $$

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UpStudy AI · Accepted Answer

Let's analyze the given functions step by step.

We need to decompose
$$
H(x) = |6x+6|
$$
into
$$
(f \circ g)(x) = f(g(x))
$$
by selecting appropriate functions $$ f $$ and $$ g $$ from the options provided.

### Option Analysis

#### Option A
- $$ f(x) = \frac{x-6}{6} $$
- $$ g(x) = | -x | = |x| $$

Then:
$$
f(g(x)) = f(|x|) = \frac{|x| - 6}{6}
$$
This does not match $$ H(x) = |6x+6| $$.

#### Option B
- $$ f(x)=|x| $$
- $$ g(x)=6x+6 $$

Then:
$$
f(g(x)) = f(6x+6) = |6x+6|
$$
This exactly matches $$ H(x) $$.

#### Option C
- $$ f(x)=|-x| = |x| $$
- $$ g(x)=\frac{x-6}{6} $$

Then:
$$
f(g(x)) = f\left(\frac{x-6}{6}\right) = \left|\frac{x-6}{6}\right|
$$
This does not equal $$ |6x+6| $$.

#### Option D
- $$ f(x)=6x+6 $$
- $$ g(x)=|x| $$

Then:
$$
f(g(x)) = f(|x|) = 6|x| + 6
$$
This does not equal $$ |6x+6| $$.

### Conclusion

The only option that results in $$ f(g(x)) = |6x+6| $$ is **Option B**.

Thus, the correct pair of functions is:

- $$ f(x)=|x| $$
- $$ g(x)=6x+6 $$
The correct pair of functions is: - $$ f(x) = |x| $$ - $$ g(x) = 6x + 6 $$

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. \( f(x)=6 x+6, g(x)=|x| \)

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