Sonia works at a bakery. The function f(x) represents the amount of money Sonia earns per loaf, where x is the number of loaves she makes. The function g(x) represents the number of bread loaves Sonia bakes per hour, where x is the number of hours she works. Show all work to find f(g(x)), and explain what f(g(x)) represents.

f(x) = 8x2 + 1
g of x equals the square root of the quantity 2 x cubed

To find \( f(g(x)) \), we need to compose the two functions. This means we will substitute the expression for \( g(x) \) into the function \( f(x) \).

### Step 1: Define the given functions
- The function for the amount of money Sonia earns per loaf is:
  \[
  f(x) = 8x^2 + 1
  \]
- The function for the number of loaves Sonia bakes per hour is:
  \[
  g(x) = \sqrt{2x^3}
  \]

### Step 2: Substitute \( g(x) \) into \( f(x) \)
We are tasked with finding \( f(g(x)) \). This involves replacing \( x \) in \( f(x) \) with \( g(x) \). Since \( g(x) = \sqrt{2x^3} \), we substitute this into \( f(x) \):

\[
f(g(x)) = f(\sqrt{2x^3}) = 8(\sqrt{2x^3})^2 + 1
\]

### Step 3: Simplify the expression
Now, simplify the expression \( 8(\sqrt{2x^3})^2 \):
- Squaring the square root cancels out the square root:
  \[
  (\sqrt{2x^3})^2 = 2x^3
  \]
- Substitute this into the equation:
  \[
  f(g(x)) = 8(2x^3) + 1
  \]
- Simplify further:
  \[
  f(g(x)) = 16x^3 + 1
  \]

### Final Expression
Thus, the composition of the functions is:
\[
f(g(x)) = 16x^3 + 1
\]

### Explanation of \( f(g(x)) \)
The function \( f(g(x)) \) represents the total amount of money Sonia earns based on the number of hours she works. Here's the breakdown:
- \( g(x) \) gives the number of loaves Sonia bakes in \( x \) hours.
- \( f(x) \) represents the amount of money Sonia earns based on the number of loaves.
- Therefore, \( f(g(x)) \) tells us how much money Sonia earns after working \( x \) hours, taking into account the number of loaves she bakes per hour and her earnings per loaf.

Would you like more details or have any questions?

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Here are 5 related questions for deeper understanding:
1. What would \( f(g(2)) \) represent, and how would you calculate it?
2. If Sonia works for 3 hours, how much money does she earn based on this model?
3. How does the rate at which Sonia bakes (i.e., \( g(x) \)) affect her total earnings?
4. What would happen to the function if \( f(x) \) were changed to \( f(x) = 10x + 2 \)?
5. How would the composition change if \( g(x) = x^2 + 2 \) instead?

**Tip**: Always simplify the inner function in function composition first before moving on to the outer function. This ensures that you apply the operations in the correct order.

Sonia works at a bakery. The function f(x) represents the amount of money Sonia earns per loaf, where x is the number of loaves she makes. The function g(x) represents the number of bread loaves Sonia bakes per hour, where x is the number of hours she works. Show all work to find f(g(x)), and explain what f(g(x)) represents. f(x) = 8x^2 + 1, g(x) = √(2x^3)

Explore the detailed solution to a function composition problem where Sonia earns money based on the number of loaves baked per hour. Learn how to calculate f(g(x)) and understand its real-world meaning using quadratic and radical functions. Suitable for Grades 10-12.

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