Let's break down the given information and find [tex]\( f(g(x)) \)[/tex] step-by-step.
### Step 1: Understand the Functions
We are given two functions:
1. [tex]\( f(x) = 2x^2 + 16 \)[/tex]
2. [tex]\( g(x) = \sqrt{5x^3} \)[/tex]
### Step 2: Interpretation of Functions
- [tex]\( f(x) \)[/tex] represents the amount of money Raul earns per ticket.
- [tex]\( g(x) \)[/tex] represents the number of tickets Raul sells per hour, where [tex]\( x \)[/tex] is the number of hours he works.
### Step 3: Composition of Functions [tex]\( f(g(x)) \)[/tex]
To find [tex]\( f(g(x)) \)[/tex], we need to substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex].
First, evaluate [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \sqrt{5x^3} \][/tex]
Now, substitute [tex]\( g(x) \)[/tex] for [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(\sqrt{5x^3}) \][/tex]
Given [tex]\( f(x) = 2x^2 + 16 \)[/tex], substitute [tex]\( \sqrt{5x^3} \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ f(\sqrt{5x^3}) = 2(\sqrt{5x^3})^2 + 16 \][/tex]
Simplify the expression inside the function:
[tex]\[ (\sqrt{5x^3})^2 = 5x^3 \][/tex]
So,
[tex]\[ f(\sqrt{5x^3}) = 2 \cdot 5x^3 + 16 \][/tex]
[tex]\[ f(\sqrt{5x^3}) = 10x^3 + 16 \][/tex]
### Step 4: Explanation of [tex]\( f(g(x)) \)[/tex]
The function [tex]\( f(g(x)) = 10x^3 + 16 \)[/tex] represents the amount of money Raul earns per hour when he sells tickets for [tex]\( x \)[/tex] hours.
### Step 5: Example Calculation for 2 Hours
To understand this better, let's compute how many tickets Raul sells and how much money he earns if he works [tex]\( x = 2 \)[/tex] hours.
1. Compute the number of tickets per hour, [tex]\( g(2) \)[/tex]:
[tex]\[ g(2) = \sqrt{5 \cdot 2^3} = \sqrt{5 \cdot 8} = \sqrt{40} \approx 6.324555320336759 \][/tex]
2. Compute the amount of money Raul earns per hour, using [tex]\( f(g(2)) \)[/tex]:
[tex]\[ f(g(2)) = 10 \cdot 2^3 + 16 = 10 \cdot 8 + 16 = 80 + 16 = 96.00000000000001 \][/tex]
So for [tex]\( x = 2 \)[/tex] hours:
- Raul sells approximately [tex]\( 6.32 \)[/tex] tickets per hour.
- He earns approximately [tex]\( \$96 \)[/tex] per hour.
This completes our detailed step-by-step breakdown and explanation of the composition of functions [tex]\( f(g(x)) \)[/tex] and what it represents.
