Sure, let's find each of the compositions step-by-step:
(a) [tex]\( f \circ g \)[/tex]:
[tex]\( f \circ g(x) \)[/tex] means we apply [tex]\( g \)[/tex] first and then apply [tex]\( f \)[/tex] to the result.
Given [tex]\( g(x) = x - 6 \)[/tex]:
1. First apply [tex]\( g \)[/tex]:
[tex]\[ g(x) = x - 6 \][/tex]
2. Now substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = f(x - 6) \][/tex]
3. Since [tex]\( f(x) = x^2 \)[/tex], replace [tex]\( x \)[/tex] with [tex]\( x - 6 \)[/tex]:
[tex]\[ f(x - 6) = (x - 6)^2 \][/tex]
So,
[tex]\[ f \circ g(x) = (x - 6)^2 \][/tex]
(b) [tex]\( g \circ f \)[/tex]:
[tex]\( g \circ f(x) \)[/tex] means we apply [tex]\( f \)[/tex] first and then apply [tex]\( g \)[/tex] to the result.
Given [tex]\( f(x) = x^2 \)[/tex]:
1. First apply [tex]\( f \)[/tex]:
[tex]\[ f(x) = x^2 \][/tex]
2. Now substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(f(x)) = g(x^2) \][/tex]
3. Since [tex]\( g(x) = x - 6 \)[/tex], replace [tex]\( x \)[/tex] with [tex]\( x^2 \)[/tex]:
[tex]\[ g(x^2) = x^2 - 6 \][/tex]
So,
[tex]\[ g \circ f(x) = x^2 - 6 \][/tex]
(c) [tex]\( g \circ g \)[/tex]:
[tex]\( g \circ g(x) \)[/tex] means we apply [tex]\( g \)[/tex] first and then apply [tex]\( g \)[/tex] again to the result.
Given [tex]\( g(x) = x - 6 \)[/tex]:
1. First apply [tex]\( g \)[/tex]:
[tex]\[ g(x) = x - 6 \][/tex]
2. Now substitute [tex]\( g(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(g(x)) = g(x - 6) \][/tex]
3. Since [tex]\( g(x) = x - 6 \)[/tex], replace [tex]\( x \)[/tex] with [tex]\( x - 6 \)[/tex]:
[tex]\[ g(x - 6) = (x - 6) - 6 = x - 12 \][/tex]
So,
[tex]\[ g \circ g(x) = x - 12 \][/tex]
Therefore, the detailed solutions are:
(a) [tex]\( f \circ g(x) = (x - 6)^2 \)[/tex]
(b) [tex]\( g \circ f(x) = x^2 - 6 \)[/tex]
(c) [tex]\( g \circ g(x) = x - 12 \)[/tex]
