Raul works at a movie theatre. the function f(x) represents the amount of money raul earns per ticket, where x is the number of tickets he sells. the function g(x) represents the number of tickets raul sells per hour, where x is the number of hours he works. show all work to find f(g( and explain what f(g(x)) represents.

f(x) = 2x2 + 16
g of x equals the square root of the quantity 5 x cubed

f(g(x)) = 10x³ + 16

f(g(x)) is the amount of money Raul earns when he works x hours

Step-by-step explanation:

* Lets explain the composite functions

- A composite function is created when one function is substituted into

 another function.

- Ex: f(g(x)) is the composite function that is formed when g(x) is

  substituted for x in f(x)

* Lets solve the problem

- The function f(x) represents the amount of money Raul earns per

  ticket, where x is the number of tickets he sells

- The function g(x) represents the number of tickets Raul sells per

  hour, where x is the number of hours he works

∵ f(x) = 2x² + 16

∵ g(x) = √(5x³)

- To find f(g(x)) substitute g(x) for x in f(x)

∴ f(g(x)) = f(√(5x³))

∴ f(g(x)) = 2[√(5x³)]² + 16

∵ [√(5x³)]² = 5x³ ⇒ power 2 canceled √

∴ f(g(x)) = 2(5x³) + 16 ⇒ simplify

∴ f(g(x)) = 10x³ + 16

∵ f(x) is the money Raul earns per ticket, where x is the number

  of tickets

∵ g(x) is the number of tickets Raul sells per hour, where x is the

   number of hours he works

∴ f(g(x)) is the amount of money Raul earns when he works x hours

* Lets explain that by numbers

- If Raul works 5 hours

∴ g(5) = √[5(5)³] = 25

∴ The number of tickets in 5 hours = 25 tickets

∴ f(25) = 2(25)² + 16

∴ f(25) = 2(625) + 16 = 1250 + 16 = 1266

∴ Raul earns $1266 in 5 hours

- If you substitute x by 5 in f(g(x)) you will have the same answer

∵ f(g(5)) = 10(5)³ + 16 = 1250 + 16 = 1266