To find the composite function [tex]\( f(g(x)) \)[/tex], we'll start by substituting the expression for [tex]\( g(x) \)[/tex] into the function [tex]\( f \)[/tex]. Let’s go through this step-by-step:
1. Functions Given:
[tex]\[
f(x) = 9x^2 + 1
\][/tex]
[tex]\[
g(x) = \sqrt{2x^3}
\][/tex]
2. Find [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = \sqrt{2x^3}
\][/tex]
This function [tex]\( g(x) \)[/tex] represents the number of bread loaves Sonia bakes per hour, where [tex]\( x \)[/tex] is the number of hours she works.
3. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
We want to find [tex]\( f(g(x)) \)[/tex]. To do this, we will substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[
f(g(x)) = f(\sqrt{2x^3})
\][/tex]
4. Evaluate [tex]\( f(\sqrt{2x^3}) \)[/tex]:
Since [tex]\( f(x) = 9x^2 + 1 \)[/tex], we need to replace [tex]\( x \)[/tex] in [tex]\( f \)[/tex] with [tex]\( \sqrt{2x^3} \)[/tex]:
[tex]\[
f(\sqrt{2x^3}) = 9(\sqrt{2x^3})^2 + 1
\][/tex]
5. Simplify the expression:
[tex]\[
(\sqrt{2x^3})^2 = 2x^3
\][/tex]
Therefore,
[tex]\[
f(\sqrt{2x^3}) = 9(2x^3) + 1 = 18x^3 + 1
\][/tex]
So, the composite function [tex]\( f(g(x)) \)[/tex] is:
[tex]\[
\boxed{18x^3 + 1}
\][/tex]
### Explanation of [tex]\( f(g(x)) \)[/tex]:
The function [tex]\( g(x) = \sqrt{2x^3} \)[/tex] represents the number of bread loaves Sonia bakes per hour.
The function [tex]\( f(x) = 9x^2 + 1 \)[/tex] represents the amount of money Sonia earns per loaf.
Therefore, the composite function [tex]\( f(g(x)) = 18x^3 + 1 \)[/tex] represents the total amount of money Sonia earns for the number of loaves she can bake in [tex]\( x \)[/tex] hours. This shows how her earnings increase based on the number of hours worked, taking into account the rate at which she bakes bread loaves and the revenue per loaf.
