Alright, let's break down the problem and solve it step-by-step:
1. Understanding the Functions:
- We are given two functions:
[tex]\[
f(x) = 2x^2 + 4
\][/tex]
[tex]\[
g(x) = \sqrt{3x^3}
\][/tex]
2. Composing the Functions:
- We need to find the composition of the functions, [tex]\( f(g(x)) \)[/tex]. This means we will substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex].
3. Substitution:
- First, determine [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = \sqrt{3x^3}
\][/tex]
- Next, substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(g(x)) = f(\sqrt{3x^3})
\][/tex]
- Using the expression for [tex]\( f(x) \)[/tex]:
[tex]\[
f(g(x)) = 2(\sqrt{3x^3})^2 + 4
\][/tex]
- Simplify the composition:
[tex]\[
(\sqrt{3x^3})^2 = 3x^3
\][/tex]
So,
[tex]\[
f(g(x)) = 2 \cdot 3x^3 + 4 = 6x^3 + 4
\][/tex]
4. Interpreting [tex]\( f(g(x)) \)[/tex]:
- The function [tex]\( g(x) = \sqrt{3x^3} \)[/tex] represents the number of gallons of ice cream Barrett makes per hour when he works for [tex]\( x \)[/tex] hours.
- The function [tex]\( f(x) = 2x^2 + 4 \)[/tex] represents the amount of money Barrett earns per gallon of ice cream.
- Therefore, [tex]\( f(g(x)) = 6x^3 + 4 \)[/tex] represents the amount of money Barrett earns per hour worked since it combines both functions: the number of gallons made per hour and the earnings per gallon.
5. Example Calculation:
- Let's calculate [tex]\( g(x) \)[/tex] and [tex]\( f(g(x)) \)[/tex] for a specific value, say [tex]\( x = 2 \)[/tex] hours:
[tex]\[
g(2) = \sqrt{3 \cdot 2^3} = \sqrt{3 \cdot 8} = \sqrt{24} = 4.898979485566356
\][/tex]
- Now compute [tex]\( f(g(2)) \)[/tex]:
[tex]\[
f(g(2)) = 6 \cdot 2^3 + 4 = 6 \cdot 8 + 4 = 48 + 4 = 52
\][/tex]
So, Barrett makes approximately 4.90 gallons of ice cream in 2 hours and earns approximately $52 for those 2 hours.
