How does one determine an organic functional group?

To determine it, you look at the molecule and see what is connected to what.  If it's connected to -OH, it's an alcohol.  C double bond O it's a ketone.  Since this is C double bond O attached to another -OR group it's an ester.

Looks like your molecule contains the -COOH group - in which case it would be a carboxylic acid - which would react with sodium hydrogen carbonate solution, giving CO2 gas (effervescence). The rest of the molecule would therefore be C6H11 (a cyclohexyl group). Of course there are many isomers of C7H12O2, esters being among them.

Yes, you have a carboxylic acid. In your expanded formula you have R-COOH. Anytime you see this, you can assume that you have a carboxylic acid.

Another quick check can be done using the condensed formula C7H12O2. Do a simple calculation for units of unsaturation using the below formula:

(2#C + #N - #halogens - #H) +2 / 2

Every 1 unit of saturation corresponds to a pi bond or a ring. (i.e.,  a double bond would have 1 unit, a triple bond would have 2 units, benzene would have four units, ect)

So in your example, you have two units of unsaturation,  indicating at least one double bond. A case such as yours where there is more that one unit of unsaturation may not be the best time to use this method.

Suppose,  however,  that you had formula C2H6O2. The units of unsaturation would be 0, indicating that you do not have a carboxylic acid or, for that matter, any carbonyl containing compound.

When in doubt, this a useful tool to fall back on.

hey ,it is easy.
remember,-
-COOH:ACID
-C=O: KETONE
-CHO:ALDEHYDE.
-CN :NITRILE.
-OH:ALCOHOL.
R-NO2:nitro alkane
and from ur formula it is carboxylic acid.