Answer :
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.
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.
