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Barrett works at an ice cream shop. The function [tex]f(x)[/tex] represents the amount of money Barrett earns per gallon of ice cream, where [tex]x[/tex] is the number of gallons of ice cream he makes. The function [tex]g(x)[/tex] represents the number of gallons of ice cream Barrett makes per hour, where [tex]x[/tex] is the number of hours he works.

Show all work to find [tex]f(g(x))[/tex], and explain what [tex]f(g(x))[/tex] represents.

[tex]\[
\begin{array}{l}
f(x) = 2x^2 + 4 \\
g(x) = \sqrt{3x^3}
\end{array}
\][/tex]


Sagot :

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.