Alright, let's carefully break down the problem step-by-step to solve for [tex]\(f(g(x))\)[/tex].
1. Understanding [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
- [tex]\(f(x) = 9x^2 + 1\)[/tex]: This function [tex]\(f(x)\)[/tex] represents the amount of money Sonia earns per loaf. Given [tex]\(x\)[/tex] as the number of loaves, [tex]\(f(x)\)[/tex] computes the earnings.
- [tex]\(g(x) = \sqrt{2x^3}\)[/tex]: This function [tex]\(g(x)\)[/tex] represents the number of loaves Sonia bakes per hour. Given [tex]\(x\)[/tex] as the number of hours she works, [tex]\(g(x)\)[/tex] computes the number of loaves baked.
2. Finding [tex]\(f(g(x))\)[/tex]:
- To compute [tex]\(f(g(x))\)[/tex], we need to substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex].
- First, let's understand what [tex]\(g(x)\)[/tex] gives us. [tex]\(g(x)\)[/tex] provides the number of loaves baked in [tex]\(x\)[/tex] hours. So, [tex]\(g(x) = \sqrt{2x^3}\)[/tex].
3. Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
- Now, we substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[
f(g(x)) = f(\sqrt{2x^3})
\][/tex]
- Recall that [tex]\(f(x)\)[/tex] is [tex]\(9x^2 + 1\)[/tex]. So wherever we see [tex]\(x\)[/tex] in [tex]\(f(x)\)[/tex], we substitute it with [tex]\(g(x)\)[/tex]:
[tex]\[
f(\sqrt{2x^3}) = 9(\sqrt{2x^3})^2 + 1
\][/tex]
4. Simplify the expression:
- Simplify the expression inside [tex]\(f(x)\)[/tex]:
[tex]\[
(\sqrt{2x^3})^2 = 2x^3
\][/tex]
- Now substitute this back into the equation:
[tex]\[
f(g(x)) = 9(2x^3) + 1
\][/tex]
- Simplify further:
[tex]\[
f(g(x)) = 18x^3 + 1
\][/tex]
5. Understanding what [tex]\(f(g(x))\)[/tex] represents:
- The function [tex]\(f(g(x))\)[/tex] represents the total earnings Sonia makes per hour she works at the bakery.
- Given [tex]\(x\)[/tex] hours of work, [tex]\(g(x)\)[/tex] gives the number of loaves baked, and [tex]\(f(g(x))\)[/tex] then gives the corresponding earnings for those loaves.
6. Verifying with example values:
- Let's look at some specific values for [tex]\(x = 0, 1, 2\)[/tex]:
- For [tex]\(x = 0\)[/tex]:
[tex]\[
f(g(0)) = 18(0)^3 + 1 = 1
\][/tex]
- For [tex]\(x = 1\)[/tex]:
[tex]\[
f(g(1)) = 18(1)^3 + 1 = 18 + 1 = 19
\][/tex]
- For [tex]\(x = 2\)[/tex]:
[tex]\[
f(g(2)) = 18(2)^3 + 1 = 18(8) + 1 = 144 + 1 = 145
\][/tex]
Combining these results, we find:
[tex]\[
\{0: 1, 1: 19, 2: 145\}
\][/tex]
This confirms:
- For 0 hours of work, Sonia earns [tex]$1.
- For 1 hour of work, Sonia earns $[/tex]19.
- For 2 hours of work, Sonia earns $145.
[tex]\(f(g(x))\)[/tex] therefore is a function that tells us Sonia's earnings based on the hours she works, efficiently combining her productivity [tex]\(g(x)\)[/tex] with her earnings per loaf [tex]\(f(x)\)[/tex].
