Questions: Find functions f and g so that f ∘ g = H. H(x)=9x+3 Choose the correct pair of functions. A. f(x)=x, g(x)=9x+3 B. f(x)=9x+3, g(x)=x C. f(x)=-x, g(x)=(x-3)/9 D. f(x)=(x-3)/9, g(x)=-x

Transcript text: Find functions f and g so that $\mathrm{f} \circ \mathrm{g}=\mathrm{H}$. \[ H(x)=|9 x+3| \] Choose the correct pair of functions. A. $f(x)=|x|, g(x)=9 x+3$ B. $f(x)=9 x+3, g(x)=|x|$ C. $f(x)=|-x|, g(x)=\frac{x-3}{9}$ D. $f(x)=\frac{x-3}{9}, g(x)=|-x|$

Solution

Solution Steps

Step 1: Identify the Given Function

The given function is $|H(x) = |9x + 3|$. This function represents the absolute value of a linear function of $x$.

Step 2: Recognize the Composition

The composition of two functions is denoted as $f(g(x))$. This means we apply $g$ first and then $f$.

Step 3: Decompose $H$ into $f$ and $g$

To decompose $H$, we set $g(x) = 9x + 3$ to match the linear part inside the absolute value of $H$. Then, we set $f(x) = |x|$ to apply the absolute value operation on the result of $g(x)$.

Step 4: Verify the Decomposition

By applying the decomposition, we verify that $f(g(x)) = |9x + 3| = H(x)$. This confirms that the decomposition correctly represents the given function $H$.