Sure! Let's go through the steps step-by-step for each part of the question.
### (a) [tex]\((f \circ g)(4)\)[/tex]
This expression means we need to evaluate [tex]\(f(g(4))\)[/tex].
1. First, evaluate [tex]\(g(4)\)[/tex]:
[tex]\[ g(x) = 9x^2 + 2 \][/tex]
Substituting [tex]\(x = 4\)[/tex]:
[tex]\[ g(4) = 9 \cdot 4^2 + 2 \][/tex]
[tex]\[ = 9 \cdot 16 + 2 \][/tex]
[tex]\[ = 144 + 2 \][/tex]
[tex]\[ = 146 \][/tex]
2. Next, evaluate [tex]\(f(g(4))\)[/tex]. We already found [tex]\(g(4) = 146\)[/tex], so we need to find [tex]\(f(146)\)[/tex]:
[tex]\[ f(x) = 3x \][/tex]
Substituting [tex]\(x = 146\)[/tex]:
[tex]\[ f(146) = 3 \cdot 146 \][/tex]
[tex]\[ = 438 \][/tex]
Therefore, [tex]\((f \circ g)(4) = 438\)[/tex].
### (b) [tex]\((g \circ f)(2)\)[/tex]
This expression means we need to evaluate [tex]\(g(f(2))\)[/tex].
1. First, evaluate [tex]\(f(2)\)[/tex]:
[tex]\[ f(x) = 3x \][/tex]
Substituting [tex]\(x = 2\)[/tex]:
[tex]\[ f(2) = 3 \cdot 2 \][/tex]
[tex]\[ = 6 \][/tex]
2. Next, evaluate [tex]\(g(f(2))\)[/tex]. We already found [tex]\(f(2) = 6\)[/tex], so we need to find [tex]\(g(6)\)[/tex]:
[tex]\[ g(x) = 9x^2 + 2 \][/tex]
Substituting [tex]\(x = 6\)[/tex]:
[tex]\[ g(6) = 9 \cdot 6^2 + 2 \][/tex]
[tex]\[ = 9 \cdot 36 + 2 \][/tex]
[tex]\[ = 324 + 2 \][/tex]
[tex]\[ = 326 \][/tex]
Therefore, [tex]\((g \circ f)(2) = 326\)[/tex].
### (c) [tex]\((f \circ f)(1)\)[/tex]
This expression means we need to evaluate [tex]\(f(f(1))\)[/tex].
1. First, evaluate [tex]\(f(1)\)[/tex]:
[tex]\[ f(x) = 3x \][/tex]
Substituting [tex]\(x = 1\)[/tex]:
[tex]\[ f(1) = 3 \cdot 1 \][/tex]
[tex]\[ = 3 \][/tex]
2. Next, evaluate [tex]\(f(f(1))\)[/tex]. We already found [tex]\(f(1) = 3\)[/tex], so we need to find [tex]\(f(3)\)[/tex]:
[tex]\[ f(x) = 3x \][/tex]
Substituting [tex]\(x = 3\)[/tex]:
[tex]\[ f(3) = 3 \cdot 3 \][/tex]
[tex]\[ = 9 \][/tex]
Therefore, [tex]\((f \circ f)(1) = 9\)[/tex].
### (d) [tex]\((g \circ g)(0)\)[/tex]
This expression means we need to evaluate [tex]\(g(g(0))\)[/tex].
1. First, evaluate [tex]\(g(0)\)[/tex]:
[tex]\[ g(x) = 9x^2 + 2 \][/tex]
Substituting [tex]\(x = 0\)[/tex]:
[tex]\[ g(0) = 9 \cdot 0^2 + 2 \][/tex]
[tex]\[ = 0 + 2 \][/tex]
[tex]\[ = 2 \][/tex]
2. Next, evaluate [tex]\(g(g(0))\)[/tex]. We already found [tex]\(g(0) = 2\)[/tex], so we need to find [tex]\(g(2)\)[/tex]:
[tex]\[ g(x) = 9x^2 + 2 \][/tex]
Substituting [tex]\(x = 2\)[/tex]:
[tex]\[ g(2) = 9 \cdot 2^2 + 2 \][/tex]
[tex]\[ = 9 \cdot 4 + 2 \][/tex]
[tex]\[ = 36 + 2 \][/tex]
[tex]\[ = 38 \][/tex]
Therefore, [tex]\((g \circ g)(0) = 38\)[/tex].
