Give a recursive function $r$ on $A$ that reverses a string
A recursive function requires base cases and recursive rules. Depending on how your system works, your base cases might include:


$r(\lambda) = \lambda$
$r(\sigma) = \sigma$ for $\sigma \in \Sigma$
etc.


You then need a general rule for more complicated cases. You're on the right track; you need a rule that takes something of the form $\sigma w$, where $\sigma \in \Sigma$ is a single letter and $w \in \Sigma^*$ is some arbitrary string, and produces the correct output. You will likely find, and should try to prove, that a rule like the following works:


$r(\sigma w) = w \sigma$


Fits the bill. Putting it all together, in pseudocode, you should arrive at something roughly like the following:

Reverse(string[1...n])
1. if n = 0 then return the empty string
2. else if n = 1 then return string[1]
3. else return the concatenation of Reverse(string[2...n]) with string[1]