To solve the given compositions of functions, we need to follow step-by-step evaluations for each part:
Given:
[tex]\[ f(x) = 5 \sqrt{x} \][/tex]
[tex]\[ g(x) = 2x \][/tex]
Let's evaluate each composition one by one:
### (a) [tex]\((f \circ g)(4)\)[/tex]
The notation [tex]\((f \circ g)(4)\)[/tex] means we first apply [tex]\(g\)[/tex] to 4, and then apply [tex]\(f\)[/tex] to the result of [tex]\(g(4)\)[/tex].
1. Calculate [tex]\(g(4)\)[/tex]:
[tex]\[ g(4) = 2 \cdot 4 = 8 \][/tex]
2. Now, apply [tex]\(f\)[/tex] to the result of [tex]\(g(4)\)[/tex]:
[tex]\[ f(8) = 5 \sqrt{8} = 5 \sqrt{4 \cdot 2} = 5 \cdot 2 \sqrt{2} = 10 \sqrt{2} \][/tex]
Thus:
[tex]\[ (f \circ g)(4) = 10\sqrt{2} \][/tex]
### (b) [tex]\((g \circ f)(2)\)[/tex]
The notation [tex]\((g \circ f)(2)\)[/tex] means we first apply [tex]\(f\)[/tex] to 2, and then apply [tex]\(g\)[/tex] to the result of [tex]\(f(2)\)[/tex].
1. Calculate [tex]\(f(2)\)[/tex]:
[tex]\[ f(2) = 5 \sqrt{2} \][/tex]
2. Now, apply [tex]\(g\)[/tex] to the result of [tex]\(f(2)\)[/tex]:
[tex]\[ g(5 \sqrt{2}) = 2 \cdot 5 \sqrt{2} = 10 \sqrt{2} \][/tex]
Thus:
[tex]\[ (g \circ f)(2) = 10\sqrt{2} \][/tex]
### (c) [tex]\((f \circ f)(1)\)[/tex]
The notation [tex]\((f \circ f)(1)\)[/tex] means we first apply [tex]\(f\)[/tex] to 1, and then apply [tex]\(f\)[/tex] to the result of [tex]\(f(1)\)[/tex].
1. Calculate [tex]\(f(1)\)[/tex]:
[tex]\[ f(1) = 5 \sqrt{1} = 5 \][/tex]
2. Now, apply [tex]\(f\)[/tex] to the result of [tex]\(f(1)\)[/tex]:
[tex]\[ f(5) = 5 \sqrt{5} \][/tex]
Thus:
[tex]\[ (f \circ f)(1) = 5\sqrt{5} \][/tex]
### (d) [tex]\((g \circ g)(0)\)[/tex]
The notation [tex]\((g \circ g)(0)\)[/tex] means we first apply [tex]\(g\)[/tex] to 0, and then apply [tex]\(g\)[/tex] to the result of [tex]\(g(0)\)[/tex].
1. Calculate [tex]\(g(0)\)[/tex]:
[tex]\[ g(0) = 2 \cdot 0 = 0 \][/tex]
2. Now, apply [tex]\(g\)[/tex] to the result of [tex]\(g(0)\)[/tex]:
[tex]\[ g(0) = 2 \cdot 0 = 0 \][/tex]
Thus:
[tex]\[ (g \circ g)(0) = 0 \][/tex]
### Final Answers:
(a) [tex]\((f \circ g)(4) = 10\sqrt{2}\)[/tex]
(b) [tex]\((g \circ f)(2) = 10\sqrt{2}\)[/tex]
(c) [tex]\((f \circ f)(1) = 5\sqrt{5}\)[/tex]
(d) [tex]\((g \circ g)(0) = 0\)[/tex]
