To find the compositions of the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex], we will work through each part step-by-step.
Given:
[tex]\[ f(x) = x + 4 \][/tex]
[tex]\[ g(x) = x - 5 \][/tex]
### (a) [tex]\( f \circ g \)[/tex]
To find [tex]\( f \circ g \)[/tex], we need to compute [tex]\( f(g(x)) \)[/tex].
1. Start with the inside function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = x - 5 \][/tex]
2. Now substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = f(x - 5) \][/tex]
3. Apply [tex]\( f(x) \)[/tex] to [tex]\( x - 5 \)[/tex]:
[tex]\[ f(x - 5) = (x - 5) + 4 \][/tex]
4. Simplify the expression:
[tex]\[ f(x - 5) = x - 1 \][/tex]
Thus, [tex]\( f \circ g = f(g(x)) = x - 1 \)[/tex].
### (b) [tex]\( g \circ f \)[/tex]
To find [tex]\( g \circ f \)[/tex], we need to compute [tex]\( g(f(x)) \)[/tex].
1. Start with the inside function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x + 4 \][/tex]
2. Now substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(f(x)) = g(x + 4) \][/tex]
3. Apply [tex]\( g(x) \)[/tex] to [tex]\( x + 4 \)[/tex]:
[tex]\[ g(x + 4) = (x + 4) - 5 \][/tex]
4. Simplify the expression:
[tex]\[ g(x + 4) = x - 1 \][/tex]
Thus, [tex]\( g \circ f = g(f(x)) = x - 1 \)[/tex].
### (c) [tex]\( g \circ g \)[/tex]
To find [tex]\( g \circ g \)[/tex], we need to compute [tex]\( g(g(x)) \)[/tex].
1. Start with the inside function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = x - 5 \][/tex]
2. Now substitute [tex]\( g(x) \)[/tex] into another [tex]\( g \)[/tex]:
[tex]\[ g(g(x)) = g(x - 5) \][/tex]
3. Apply [tex]\( g(x) \)[/tex] to [tex]\( x - 5 \)[/tex]:
[tex]\[ g(x - 5) = (x - 5) - 5 \][/tex]
4. Simplify the expression:
[tex]\[ g(x - 5) = x - 10 \][/tex]
Thus, [tex]\( g \circ g = g(g(x)) = x - 10 \)[/tex].
So the final results are:
(a) [tex]\( f \circ g = x - 1 \)[/tex]
(b) [tex]\( g \circ f = x - 1 \)[/tex]
(c) [tex]\( g \circ g = x - 10 \)[/tex]
