Make a number palindrome

Perl, 32 chars

s/((.)(?1)\2|.?)$/$&.reverse$`/e

Needs Perl 5.10 or later for regex features, but no special command-line switch.

Sample use:

$ perl -pe 's/((.)(?1)\2|.?)$/$&.reverse$`/e' << EOT
> 12
> 232
> 2323
> 1012121
> EOT
121
232
23232
101212101

Uses Perl 5.10's recursive regex extensions to match the longest trailing palindrome as such:

m/
    (      # paren 1 - a palindrome is either:
      (.)  # paren 2 - a character
      (?1) # a palindrome as defined in paren 1
      \2   # the same character as in paren 2
    |        # or:
      .?   # a 0- or 1-character string
    )
    $      # at end of string
/x

It then replaces it with itself ($&) and appends whatever the string started with ($`), reversed.

Brachylog 2, 8 bytes, language postdates challenge

ẹ;AcB↔Bc

Try it online! The question asks for a function, so I provided one; the TIO link takes an argument that runs a function like a full program.

Explanation

ẹ;AcB↔Bc
ẹ          Split {the input} into digits
 ;Ac       Append {the shortest possible} list
    B↔B    to produce a palindrome
       c   then concatenate the resulting list of digits back into a number

J, ^{50, 32} 26 characters!

f=:{.@(,"1(-:|.)\.#|.@}:\)

eg

f '12'
121 
f '232'
232   
f '2323'
23232   
f '1012121'
101212101     

How it works (by example)

y =: '1012121'

[\.y   NB. Sub lists of y
1012121
012121 
12121  
2121   
121    
21     
1 

|.\. y  NB> Reverses of sub lists of y
1212101
121210 
12121  
1212   
121    
12     
1  

([\. y) -:"1 (|. \. y) NB. Which of them are equal? (those are palindromes)
                       NB. ( -:"1 ) checks equality item by item
0 0 1 0 1 0 1

(-:  |.)\. y NB. Shortcut of the above
0 0 1 0 1 0 1

(0 0 1 0 1 0 1) # }:\y NB. Choose (#) the palindrome prefixes (\)
10    
1012  
101212

y, |.'10'   NB. Reverse and append the first prefix.
101212101