To solve the given expressions using the functions [tex]\(f(x) = 5x\)[/tex] and [tex]\(g(x) = 7x^2 + 3\)[/tex], we need to go through each subpart step-by-step. Let's evaluate each expression carefully:
### (a) [tex]\((f \circ g)(4)\)[/tex]
The notation [tex]\((f \circ g)(4)\)[/tex] means we need to apply [tex]\(g(x)\)[/tex] first and then apply [tex]\(f(x)\)[/tex] to the result.
1. First, compute [tex]\(g(4)\)[/tex]:
[tex]\[ g(4) = 7(4)^2 + 3 \][/tex]
Simplifying inside the parenthesis first:
[tex]\[ = 7 \cdot 16 + 3 \][/tex]
[tex]\[ = 112 + 3 \][/tex]
[tex]\[ = 115 \][/tex]
2. Next, use this result in [tex]\(f(x)\)[/tex]:
[tex]\[ f(g(4)) = f(115) \][/tex]
[tex]\[ = 5 \cdot 115 \][/tex]
[tex]\[ = 575 \][/tex]
Thus, [tex]\((f \circ g)(4) = 575\)[/tex].
### (b) [tex]\((g \circ f)(2)\)[/tex]
The notation [tex]\((g \circ f)(2)\)[/tex] means we need to apply [tex]\(f(x)\)[/tex] first and then apply [tex]\(g(x)\)[/tex] to the result.
1. First, compute [tex]\(f(2)\)[/tex]:
[tex]\[ f(2) = 5 \cdot 2 \][/tex]
[tex]\[ = 10 \][/tex]
2. Next, use this result in [tex]\(g(x)\)[/tex]:
[tex]\[ g(f(2)) = g(10) \][/tex]
[tex]\[ = 7 \cdot 10^2 + 3 \][/tex]
Simplify inside the parenthesis first:
[tex]\[ = 7 \cdot 100 + 3 \][/tex]
[tex]\[ = 700 + 3 \][/tex]
[tex]\[ = 703 \][/tex]
Thus, [tex]\((g \circ f)(2) = 703\)[/tex].
### (c) [tex]\((f \circ f)(1)\)[/tex]
The notation [tex]\((f \circ f)(1)\)[/tex] means we need to apply [tex]\(f(x)\)[/tex] twice.
1. First, compute [tex]\(f(1)\)[/tex]:
[tex]\[ f(1) = 5 \cdot 1 \][/tex]
[tex]\[ = 5 \][/tex]
2. Next, use this result in [tex]\(f(x)\)[/tex] again:
[tex]\[ f(f(1)) = f(5) \][/tex]
[tex]\[ = 5 \cdot 5 \][/tex]
[tex]\[ = 25 \][/tex]
Thus, [tex]\((f \circ f)(1) = 25\)[/tex].
### (d) [tex]\((g \circ g)(0)\)[/tex]
The notation [tex]\((g \circ g)(0)\)[/tex] means we need to apply [tex]\(g(x)\)[/tex] twice.
1. First, compute [tex]\(g(0)\)[/tex]:
[tex]\[ g(0) = 7 \cdot 0^2 + 3 \][/tex]
[tex]\[ = 0 + 3 \][/tex]
[tex]\[ = 3 \][/tex]
2. Next, use this result in [tex]\(g(x)\)[/tex] again:
[tex]\[ g(g(0)) = g(3) \][/tex]
[tex]\[ = 7 \cdot 3^2 + 3 \][/tex]
Simplify inside the parenthesis first:
[tex]\[ = 7 \cdot 9 + 3 \][/tex]
[tex]\[ = 63 + 3 \][/tex]
[tex]\[ = 66 \][/tex]
Thus, [tex]\((g \circ g)(0) = 66\)[/tex].
### Summary of Solutions:
(a) [tex]\((f \circ g)(4) = 575\)[/tex]
(b) [tex]\((g \circ f)(2) = 703\)[/tex]
(c) [tex]\((f \circ f)(1) = 25\)[/tex]
(d) [tex]\((g \circ g)(0) = 66\)[/tex]
