Question

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.

Answer 1

#{: ("|",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`