Complete the table given h(x)=g(f(x)) ?

I'm really confused on how to use composition of functions to complete the table. I was only able to fill in one blank. I would much appreciate some help with steps on how to solve this table.enter image source here

1 Answer
Jun 28, 2018

#{: ("|",ul(x),"||",ul(f(x)),"|",ul(g(x)),"|",ul(h(x)=g(f(x))),"|"), ("|",0,"||",2,"|",color(red)1,"|",color(red)3,"|"), ("|",1,"||",color(red)1,"|",0,"|",0,"|"),("|",2,"||",color(red)4,"|",3,"|",color(red)2,"|"),("|",3,"||",0,"|",color(red)4,"|",1,"|"), ("|",4,"||",3,"|",2,"|",4,"|") :}#

Explanation:

Given (with identification variables added for later reference):
#{: ("|",ul(x),"||",ul(f(x)),"|",ul(g(x)),"|",ul(h(x)=g(f(x))),"|"), ("|",0,"||",2,"|",color(red)(ul(" A ")),"|",color(red)(ul(" B ")),"|"), ("|",1,"||",color(red)(ul(" C ")),"|",0,"|",0,"|"),("|",2,"||",color(red)(ul(" D ")),"|",3,"|",color(red)(ul(" E ")),"|"),("|",3,"||",0,"|",color(red)(ul(" F ")),"|",1,"|"), ("|",4,"||",3,"|",2,"|",4,"|") :}#

#h(0)=g(f(0))=g(2)=3#
#color(red)B=color(red)3#

#h(4)=g(f(4))=g(3)#
but we are also told that #h(4)=4# so #g(3)=4#
#color(red)"F"=color(red)4#

#h(3)=g(f(3))=g(0)#
#color(red)"A"=color(red)1#
but we are also told that #h(3)=1# so #g(0)=1#

From here on, I am not certain that any unique solution is possible #ul("unless")# we make some assumptions.

I have assumed that the functions are one-to-one and the range is limited to #{0,1,2,3,4}#

If this is the case, the only value remaining for #h(x)# is
#color(red)("E")=color(red)2#
and
since #g(4)=2# and #g(f(2))=2#
#f(2)=4# ...if one-to-one functions
#color(red)"D"=color(red)4#

and this only leaves
#color(red)C=color(red)1#