To find [tex]\( g \circ f \)[/tex] (denoted as [tex]\( gof \)[/tex]), we need to determine the composition of the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex]. The composition [tex]\( gof \)[/tex] is defined such that for each [tex]\( x \)[/tex] in the domain of [tex]\( f \)[/tex], we find [tex]\( f(x) \)[/tex], and then apply [tex]\( g \)[/tex] to the result of [tex]\( f(x) \)[/tex].
We start with the function [tex]\( f \)[/tex] given as:
[tex]\[ f = \{(1, 2), (3, 5), (4, 1)\} \][/tex]
And the function [tex]\( g \)[/tex] given as:
[tex]\[ g = \{(2, 3), (5, 1), (1, 6)\} \][/tex]
Let's evaluate step by step:
1. For the pair [tex]\( (1, 2) \)[/tex] in [tex]\( f \)[/tex]:
- [tex]\( f(1) = 2 \)[/tex]
- Now, find [tex]\( g(2) \)[/tex]:
[tex]\[ g(2) = 3 \][/tex]
- So, for [tex]\( x = 1 \)[/tex], [tex]\( g(f(1)) = g(2) = 3 \)[/tex]
- Thus, one element in [tex]\( gof \)[/tex] is [tex]\( (1, 3) \)[/tex]
2. For the pair [tex]\( (3, 5) \)[/tex] in [tex]\( f \)[/tex]:
- [tex]\( f(3) = 5 \)[/tex]
- Now, find [tex]\( g(5) \)[/tex]:
[tex]\[ g(5) = 1 \][/tex]
- So, for [tex]\( x = 3 \)[/tex], [tex]\( g(f(3)) = g(5) = 1 \)[/tex]
- Thus, another element in [tex]\( gof \)[/tex] is [tex]\( (3, 1) \)[/tex]
3. For the pair [tex]\( (4, 1) \)[/tex] in [tex]\( f \)[/tex]:
- [tex]\( f(4) = 1 \)[/tex]
- Now, find [tex]\( g(1) \)[/tex]:
[tex]\[ g(1) = 6 \][/tex]
- So, for [tex]\( x = 4 \)[/tex], [tex]\( g(f(4)) = g(1) = 6 \)[/tex]
- Thus, another element in [tex]\( gof \)[/tex] is [tex]\( (4, 6) \)[/tex]
Combining all the elements we found, we get:
[tex]\[ gof = \{(1, 3), (3, 1), (4, 6)\} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{\{(1, 3), (3, 1), (4, 6)\}} \][/tex]
