Question 1200475
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Answer: <font color=red size=4>(f o g)(x) = {(8,3), (-1,8), (2,-1)}</font>


Explanation:


(f o g)(x) is the same as f( g(x) )


The g(x) function is
g(x) = {(8,2), (-1,4), (2,7)}
which means
g(8) = 2
g(-1) = 4
g(2) = 7
Each of those outputs (2,4 and 7) must be an input for the f(x) function to have f(g(x)) be possible.
Luckily that is indeed the case.


f(x) = {(2,3), (4,8), (7,-1)}
breaks down to
f(2) = 3
f(4) = 8
f(7) = -1



Let's rewrite those three items above to involve g(x)
f(2) = 3 ---> f(g(8)) = 3
f(4) = 8 ---> f(g(-1)) = 8
f(7) = -1 ---> f(g(2)) = -1


In short,
f(g(8)) = 3
f(g(-1)) = 8
f(g(2)) = -1
or
(f o g)(8) = 3
(f o g)(-1) = 8
(f o g)(2) = -1
Therefore,
<font color=red size=4>(f o g)(x) = {(8,3), (-1,8), (2,-1)}</font>


Here's the input/output mapping diagram for each function piece. 
Start at the left side and move toward the right along the pathway of arrows shown.
{{{
drawing(300,300,-5,3,-4,4,
locate(-3,2.5,"x"),
locate(-3,1,"8"),
locate(-3-0.1,0,"-1"),
locate(-3,-1,"2"),
ellipse(-3+0.1,0-0.1,1,4),


locate(-1-0.3,2.5,"g(x)"),
locate(-1,1,"2"),
locate(-1,0,"4"),
locate(-1,-1,"7"),
ellipse(-1+0.1,0-0.1,1,4),

locate(1-0.2,3.5,"f(g(x))"),
locate(1,3,"aka"),
locate(1-0.3,2.5,"(f o g)(x)"),
locate(1,1,"3"),
locate(1,0,"8"),
locate(1-0.1,-1,"-1"),
ellipse(1+0.1,0-0.1,1,4),

line(-2.3,0.8+0,-1.5,0.8+0),
line(-1.5,0.8+0,-1.7,0.95+0),
line(-1.5,0.8+0,-1.7,0.65+0),

line(-2.3+2,0.8+0,-1.5+2,0.8+0),
line(-1.5+2,0.8+0,-1.7+2,0.95+0),
line(-1.5+2,0.8+0,-1.7+2,0.65+0),


line(-2.3,0.8+-1,-1.5,0.8+-1),
line(-1.5,0.8+-1,-1.7,0.95+-1),
line(-1.5,0.8+-1,-1.7,0.65+-1),

line(-2.3+2,0.8+-1,-1.5+2,0.8+-1),
line(-1.5+2,0.8+-1,-1.7+2,0.95+-1),
line(-1.5+2,0.8+-1,-1.7+2,0.65+-1),


line(-2.3,0.8+-2,-1.5,0.8+-2),
line(-1.5,0.8+-2,-1.7,0.95+-2),
line(-1.5,0.8+-2,-1.7,0.65+-2),

line(-2.3+2,0.8+-2,-1.5+2,0.8+-2),
line(-1.5+2,0.8+-2,-1.7+2,0.95+-2),
line(-1.5+2,0.8+-2,-1.7+2,0.65+-2)
)
}}}
For instance, if the input is x = 8, then
g(8) = 2
f(2) = 3 aka f(g(8)) = 3
Therefore, f(g(x)) = 3 when the input is x = 8.


The g(x) function acts as the input to the f(x) function.
Effectively g(x) takes on the role of x so to speak.
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