To find two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] such that [tex]\( (f \circ g)(x) = H(x) \)[/tex], where [tex]\( H(x) = \sqrt[3]{3 x^2 + 8} \)[/tex], we need to decompose [tex]\( H(x) \)[/tex] into the composition of two functions.
### Step-by-Step Solution
1. Identify the Inner Function [tex]\( g(x) \)[/tex]:
We need to find a function [tex]\( g(x) \)[/tex] whose output we can plug into another function [tex]\( f \)[/tex] to yield [tex]\( H(x) \)[/tex]. Let's set our inner function to be:
[tex]\[
g(x) = 3x^2 + 8
\][/tex]
2. Find the Outer Function [tex]\( f(x) \)[/tex]:
Next, we need the outer function [tex]\( f \)[/tex] that will take [tex]\( g(x) \)[/tex] as its input. The goal is that [tex]\( f(g(x)) \)[/tex] should be [tex]\( H(x) \)[/tex]. Since:
[tex]\[
H(x) = \sqrt[3]{3x^2 + 8}
\][/tex]
And we have [tex]\( g(x) = 3x^2 + 8 \)[/tex], we see that:
[tex]\[
H(x) = \sqrt[3]{g(x)}
\][/tex]
Hence, we can define:
[tex]\[
f(x) = x^{1/3}
\][/tex]
### Verification
- Calculating [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = 3x^2 + 8
\][/tex]
- Calculating [tex]\( f(g(x)) \)[/tex]:
[tex]\[
f(g(x)) = f(3x^2 + 8) = (3x^2 + 8)^{1/3}
\][/tex]
- Compare with [tex]\( H(x) \)[/tex]:
[tex]\[
H(x) = \sqrt[3]{3x^2 + 8}
\][/tex]
We observe that:
[tex]\[
f(g(x)) = (3x^2 + 8)^{1/3} = \sqrt[3]{3x^2+8} = H(x)
\][/tex]
Hence, we confirm that the functions [tex]\( f(x) = x^{1/3} \)[/tex] and [tex]\( g(x) = 3x^2 + 8 \)[/tex] satisfy:
[tex]\[
(f \circ g)(x) = H(x)
\][/tex]
Thus, the solution is:
[tex]\[
f(x) = x^{1/3}, \quad g(x) = 3x^2 + 8
\][/tex]
