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)=2 x^2+4 \\
g(x)=\sqrt{3 x^3}
\end{array}
[/tex]
Answer (2)
We found that f ( g ( x )) = 6 x 3 + 4 , which represents the amount of money Barrett earns after working x hours. This is calculated by substituting g ( x ) into f ( x ) , leading to the total earnings depending on the gallons of ice cream produced per hour. This relationship helps Barrett understand his earnings based on his working hours.
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Find the composite function f ( g ( x )) by substituting g ( x ) into f ( x ) .
Substitute g ( x ) = 3 x 3 into f ( x ) = 2 x 2 + 4 , which gives f ( g ( x )) = 2 ( 3 x 3 ) 2 + 4 .
Simplify the expression to get f ( g ( x )) = 6 x 3 + 4 .
f ( g ( x )) = 6 x 3 + 4 represents the amount of money Barrett earns when he works for x hours. f ( g ( x )) = 6 x 3 + 4
Explanation
Understanding the Problem We are given two functions: f ( x ) = 2 x 2 + 4 and g ( x ) = 3 x 3 . We need to find the composite function f ( g ( x )) and explain what it represents in the context of the problem.
Finding the Composite Function To find f ( g ( x )) , we need to substitute g ( x ) into f ( x ) . This means replacing every instance of x in the function f ( x ) with the entire function g ( x ) .
Substitution So, we have f ( g ( x )) = 2 ( g ( x ) ) 2 + 4 . Now, substitute g ( x ) = 3 x 3 : f ( g ( x )) = 2 ( 3 x 3 ) 2 + 4
Simplification Next, we simplify the expression: f ( g ( x )) = 2 ( 3 x 3 ) + 4 f ( g ( x )) = 6 x 3 + 4
Interpretation Now, let's interpret what f ( g ( x )) represents. We know that g ( x ) represents the number of gallons of ice cream Barrett makes per hour, where x is the number of hours he works. Also, f ( x ) represents the amount of money Barrett earns per gallon of ice cream, where x is the number of gallons of ice cream he makes. Therefore, f ( g ( x )) represents the amount of money Barrett earns, where x is the number of hours he works. In other words, it tells us how much money Barrett makes based on the number of hours he works.
Final Answer Therefore, f ( g ( x )) = 6 x 3 + 4 represents the amount of money Barrett earns when he works for x hours.
Examples
Imagine you're planning a summer job at an ice cream shop. Knowing how your earnings relate to the hours you work can help you set financial goals. By understanding composite functions like f ( g ( x )) , you can predict your income based on the number of hours worked. For example, if you want to earn a certain amount, you can calculate how many hours you need to work. This kind of math is useful for budgeting and making informed decisions about your time and money.
