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If [tex][tex]$f = \{(1,2), (3,5), (4,1)\}$[/tex][/tex] and [tex][tex]$g = \{(2,3), (5,1), (1,6)\}$[/tex][/tex], then [tex][tex]$g \circ f$[/tex][/tex] is equal to:

(a) [tex][tex]$\{(2,1), (1,5), (6,4)\}$[/tex][/tex]
(b) [tex][tex]$\{(1,3) \cdot (3,1) \cdot (4,6)\}$[/tex][/tex]
(c) [tex][tex]$\{(1,5), (3,2), (4,6)\}$[/tex][/tex]
(d) __

(Note: The last option (d) appears incomplete. Either remove it if no other option is available or provide the correct missing option if necessary.)



Answer :

GinnyAnswer
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]

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