You have a major error in that you are using the range operator which will not work correctly, for example:

$ perl -le'my @x = $ARGV[1] .. $ARGV[$#ARGV]; print "@x"' add  12  8  
+2

$ perl -le'my @x = $ARGV[1] .. $ARGV[$#ARGV]; print "@x"' add  2  8  1
+2
2 3 4 5 6 7 8 9 10 11 12
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You should probably shift the first argument off @ARGV, something like this:

use warnings;
use strict;

my $i = shift;

if ( $i eq 'add' ) {
    my $rtn = add( @ARGV );
    print "The sum is: $rtn";
}
elsif ( $i eq 'multiply' ){
    my $rtn = multiply( @ARGV );
    print "Their product is: $rtn";
}

sub add {
    my $sum = 0;
    foreach ( @_ ) {
        $sum += $_;
    }
    return $sum;
}

sub multiply {
    my $product = 1;
    foreach ( @_ ) {
        $product *= $_;
    }
    return $product;
}
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