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.