Answer :
To solve these composition problems involving the functions [tex]\( f(x) = 3x^2 - 2 \)[/tex] and [tex]\( g(x) = 7 - \frac{1}{2}x^2 \)[/tex], we need to evaluate each expression step-by-step. Let’s tackle each part independently.
### (a) [tex]\((f \circ g)(4)\)[/tex]
The notation [tex]\((f \circ g)(4)\)[/tex] means [tex]\(f(g(4))\)[/tex]. We first find [tex]\(g(4)\)[/tex] and then apply [tex]\(f\)[/tex] to that result.
1. Calculate [tex]\( g(4) \)[/tex]:
[tex]\[ g(4) = 7 - \frac{1}{2}(4)^2 = 7 - \frac{1}{2}(16) = 7 - 8 = -1 \][/tex]
2. Now calculate [tex]\( f(g(4)) = f(-1) \)[/tex]:
[tex]\[ f(-1) = 3(-1)^2 - 2 = 3(1) - 2 = 3 - 2 = 1 \][/tex]
So, [tex]\((f \circ g)(4) = 1\)[/tex].
### (b) [tex]\((g \circ f)(2) \)[/tex]
The notation [tex]\((g \circ f)(2)\)[/tex] means [tex]\(g(f(2))\)[/tex]. We first find [tex]\(f(2)\)[/tex] and then apply [tex]\(g\)[/tex] to that result.
1. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 3(2)^2 - 2 = 3(4) - 2 = 12 - 2 = 10 \][/tex]
2. Now calculate [tex]\( g(f(2)) = g(10) \)[/tex]:
[tex]\[ g(10) = 7 - \frac{1}{2}(10)^2 = 7 - \frac{1}{2}(100) = 7 - 50 = -43 \][/tex]
So, [tex]\((g \circ f)(2) = -43\)[/tex].
### (c) [tex]\((f \circ f)(1) \)[/tex]
The notation [tex]\((f \circ f)(1)\)[/tex] means [tex]\(f(f(1))\)[/tex]. We first find [tex]\(f(1)\)[/tex] and then apply [tex]\(f\)[/tex] to that result.
1. Calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 3(1)^2 - 2 = 3(1) - 2 = 3 - 2 = 1 \][/tex]
2. Now calculate [tex]\( f(f(1)) = f(1) \)[/tex] (Note that in this case, since [tex]\(f(1) = 1\)[/tex], it’s the same computation again.):
[tex]\[ f(1) = 3(1)^2 - 2 = 3(1) - 2 = 3 - 2 = 1 \][/tex]
So, [tex]\((f \circ f)(1) = 1\)[/tex].
### (d) [tex]\((g \circ g)(0) \)[/tex]
The notation [tex]\((g \circ g)(0)\)[/tex] means [tex]\(g(g(0))\)[/tex]. We first find [tex]\(g(0)\)[/tex] and then apply [tex]\(g\)[/tex] to that result.
1. Calculate [tex]\( g(0) \)[/tex]:
[tex]\[ g(0) = 7 - \frac{1}{2}(0)^2 = 7 - \frac{1}{2}(0) = 7 - 0 = 7 \][/tex]
2. Now calculate [tex]\( g(g(0)) = g(7) \)[/tex]:
[tex]\[ g(7) = 7 - \frac{1}{2}(7)^2 = 7 - \frac{1}{2}(49) = 7 - 24.5 = -17.5 \][/tex]
So, [tex]\((g \circ g)(0) = -17.5\)[/tex].
### Summary:
(a) [tex]\((f \circ g)(4) = 1\)[/tex]
(b) [tex]\((g \circ f)(2) = -43\)[/tex]
(c) [tex]\((f \circ f)(1) = 1\)[/tex]
(d) [tex]\((g \circ g)(0) = -17.5\)[/tex]
### (a) [tex]\((f \circ g)(4)\)[/tex]
The notation [tex]\((f \circ g)(4)\)[/tex] means [tex]\(f(g(4))\)[/tex]. We first find [tex]\(g(4)\)[/tex] and then apply [tex]\(f\)[/tex] to that result.
1. Calculate [tex]\( g(4) \)[/tex]:
[tex]\[ g(4) = 7 - \frac{1}{2}(4)^2 = 7 - \frac{1}{2}(16) = 7 - 8 = -1 \][/tex]
2. Now calculate [tex]\( f(g(4)) = f(-1) \)[/tex]:
[tex]\[ f(-1) = 3(-1)^2 - 2 = 3(1) - 2 = 3 - 2 = 1 \][/tex]
So, [tex]\((f \circ g)(4) = 1\)[/tex].
### (b) [tex]\((g \circ f)(2) \)[/tex]
The notation [tex]\((g \circ f)(2)\)[/tex] means [tex]\(g(f(2))\)[/tex]. We first find [tex]\(f(2)\)[/tex] and then apply [tex]\(g\)[/tex] to that result.
1. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 3(2)^2 - 2 = 3(4) - 2 = 12 - 2 = 10 \][/tex]
2. Now calculate [tex]\( g(f(2)) = g(10) \)[/tex]:
[tex]\[ g(10) = 7 - \frac{1}{2}(10)^2 = 7 - \frac{1}{2}(100) = 7 - 50 = -43 \][/tex]
So, [tex]\((g \circ f)(2) = -43\)[/tex].
### (c) [tex]\((f \circ f)(1) \)[/tex]
The notation [tex]\((f \circ f)(1)\)[/tex] means [tex]\(f(f(1))\)[/tex]. We first find [tex]\(f(1)\)[/tex] and then apply [tex]\(f\)[/tex] to that result.
1. Calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 3(1)^2 - 2 = 3(1) - 2 = 3 - 2 = 1 \][/tex]
2. Now calculate [tex]\( f(f(1)) = f(1) \)[/tex] (Note that in this case, since [tex]\(f(1) = 1\)[/tex], it’s the same computation again.):
[tex]\[ f(1) = 3(1)^2 - 2 = 3(1) - 2 = 3 - 2 = 1 \][/tex]
So, [tex]\((f \circ f)(1) = 1\)[/tex].
### (d) [tex]\((g \circ g)(0) \)[/tex]
The notation [tex]\((g \circ g)(0)\)[/tex] means [tex]\(g(g(0))\)[/tex]. We first find [tex]\(g(0)\)[/tex] and then apply [tex]\(g\)[/tex] to that result.
1. Calculate [tex]\( g(0) \)[/tex]:
[tex]\[ g(0) = 7 - \frac{1}{2}(0)^2 = 7 - \frac{1}{2}(0) = 7 - 0 = 7 \][/tex]
2. Now calculate [tex]\( g(g(0)) = g(7) \)[/tex]:
[tex]\[ g(7) = 7 - \frac{1}{2}(7)^2 = 7 - \frac{1}{2}(49) = 7 - 24.5 = -17.5 \][/tex]
So, [tex]\((g \circ g)(0) = -17.5\)[/tex].
### Summary:
(a) [tex]\((f \circ g)(4) = 1\)[/tex]
(b) [tex]\((g \circ f)(2) = -43\)[/tex]
(c) [tex]\((f \circ f)(1) = 1\)[/tex]
(d) [tex]\((g \circ g)(0) = -17.5\)[/tex]