Sure, let's go through solving each of these expressions step by step.
Given the functions:
[tex]\[ f(x) = 3\sqrt{x} \][/tex]
[tex]\[ g(x) = 6x \][/tex]
We need to find the following compositions:
### (a) [tex]\((f \circ g)(4)\)[/tex]
This means we need to find [tex]\( f(g(4)) \)[/tex].
1. Calculate [tex]\( g(4) \)[/tex]:
[tex]\[ g(4) = 6 \times 4 = 24 \][/tex]
2. Plug [tex]\( g(4) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(24) = 3 \sqrt{24} \][/tex]
[tex]\[ \sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6} \][/tex]
[tex]\[ f(24) = 3 \times 2\sqrt{6} = 6\sqrt{6} \][/tex]
So, [tex]\((f \circ g)(4) = 6\sqrt{6}\)[/tex].
### (b) [tex]\((g \circ f)(2)\)[/tex]
This means we need to find [tex]\( g(f(2)) \)[/tex].
1. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 3\sqrt{2} \][/tex]
2. Plug [tex]\( f(2) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(3\sqrt{2}) = 6 \times 3\sqrt{2} = 18\sqrt{2} \][/tex]
So, [tex]\((g \circ f)(2) = 18\sqrt{2}\)[/tex].
### (c) [tex]\((f \circ f)(1)\)[/tex]
This means we need to find [tex]\( f(f(1)) \)[/tex].
1. Calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 3\sqrt{1} = 3 \][/tex]
2. Plug [tex]\( f(1) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(3) = 3\sqrt{3} \][/tex]
So, [tex]\((f \circ f)(1) = 3\sqrt{3}\)[/tex].
### (d) [tex]\((g \circ g)(0)\)[/tex]
This means we need to find [tex]\( g(g(0)) \)[/tex].
1. Calculate [tex]\( g(0) \)[/tex]:
[tex]\[ g(0) = 6 \times 0 = 0 \][/tex]
2. Plug [tex]\( g(0) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(0) = 6 \times 0 = 0 \][/tex]
So, [tex]\((g \circ g)(0) = 0\)[/tex].
To wrap it up, the exact answers for the given compositions are:
(a) [tex]\( (f \circ g)(4) = 6\sqrt{6} \)[/tex]
(b) [tex]\( (g \circ f)(2) = 18\sqrt{2} \)[/tex]
(c) [tex]\( (f \circ f)(1) = 3\sqrt{3} \)[/tex]
(d) [tex]\( (g \circ g)(0) = 0 \)[/tex]
