Step 1: Understand the definition of a functional group.
A functional group is a specific group of atoms within a molecule that is responsible for the characteristic chemical reactions of that molecule.
Step 2: Recall the common functional groups and their formulas.
Aldehyde group: -CHO (or -CH=O)
Carboxylic acid group: -COOH (or -C(=O)OH)
Ketone group: -C(=O)- (often shown within a molecule like -COCH$_3$ for a methyl ketone, but the group itself is just the carbonyl carbon bonded to two other carbons)
Hydroxyl group (Alcohol): -OH
Step 3: Identify the formula for the Carboxylic group.
Based on the recall, the carboxylic group is characterized by a carbon atom double-bonded to an oxygen atom and single-bonded to a hydroxyl (-OH) group. Its formula is -COOH.
Step 4: Compare the conclusion with the given options.
Option (2) -COOH correctly represents the carboxylic group.
$$(2) -COOH$$