Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
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.
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.
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.