Question

Consider the following recursive function: int mystery (int number) / / Line 1 \(f\) If (number = 0) / / Line 2 return number; else return (mystery (number +1 ) \(-\) number) \(;\) //Line 5\\} a. Identify the base case. b. Identify the general case. c. What valid values can be passed as parameters to the function mystery? d. If mystery (0) is a valid call, what is its value? If not, explain why. e. If mystery (10) is a valid call, what is its value? If not, explain why. f. If mystery (-3) is a valid call, what is its value? If not, explain why.

Short answer

a) Base case: number = 0. b) General case: mystery(number + 1) - number. c) Any integer. d) 0; e) -55; f) -6.

Step-by-step solution

Identify the Base Case

The base case occurs when the recursive function stops making recursive calls. Look at the condition given on Line 2: `if (number == 0) return number;`. Hence, the base case is when `number` equals 0, and the function returns 0.

Identify the General Case

The general case describes how the recursion progresses. From the function, if `number != 0`, it recursively calls itself with `mystery(number + 1)` and returns the value of that call minus `number`. This is the recursive step or the general case.

Valid Parameter Values

For any recursive function to terminate, it must eventually reach the base case. For this function, starting with any integer will eventually reach 0 through repeated increments by 1. Therefore, all integers are valid inputs to the function.

Evaluate mystery(0)

The call `mystery(0)` directly hits the base case at Line 2. According to the base case, the function will return 0. Thus, `mystery(0)` is valid and its return value is `0`.

Evaluate mystery(10)

To evaluate `mystery(10)`, observe how recursion unfolds: `10 -> 11 -> 12 -> ... -> 0`. At each step, the recursion returns `mystery(number + 1) - number` until reaching 0. Eventually, due to progressive increments, `mystery(0)` returns 0, which will backtrack through the recursive calls. The returns subtract progressively larger passed numbers, converging to -55. Thus, `mystery(10)` is valid and the return value is `-55`.

Evaluate mystery(-3)

Recursively, `-3 -> -2 -> -1 -> 0`. Travel upward through recursive calls, reaching the base case `mystery(0)` which returns 0, and then backtracks. The computed returns deductively lead to cancelation out across descending integers. Upon full stack unwinding, `mystery(-3)` yields a return value of `-6`. Hence, `mystery(-3)` is valid with a result of `-6`.

Key concepts

Base Case

In recursive functions, the base case is crucial as it defines the condition under which the recursion stops. Without a base case, the recursion could continue indefinitely. In the given function, the base case is defined in Line 2 with the condition `if (number == 0) return number;`. This means when the `number` equals zero, the function stops calling itself and returns 0. The base case ensures the recursive function has a clear endpoint, thereby preventing infinite loops and potentially crashing the program.

General Case

The general case in a recursive function outlines how the function progresses through its recursive steps. It shows the logic applied to reach closer to the base case. In our function, when the number is not zero, it follows the logic shown in Line 5: `return (mystery(number + 1) - number);`. This step is responsible for pushing the function closer to the base case by incrementing the number by 1 and making another recursive call. This process continues until the base case is reached at `number == 0`. The general case is the core of recursion as it determines how each step is computed based on the previous one.

Function Parameters

Function parameters are the inputs that a function takes to perform its computation or operation. In recursive functions, choosing the right parameters is important to ensure the function will eventually terminate. For the `mystery` function, any integer value can be passed as a parameter. Whether positive or negative, each call incrementally increases the `number` until it reaches zero. Hence, all integers are valid parameters for this function. Selecting suitable parameters helps the function to reach the base case without errors and ensures the correct functioning of the recursive algorithm.

Recursion

Recursion is a method where a function calls itself with modified parameters to solve a problem. It's a powerful technique for solving problems that can be broken down into smaller, similar sub-problems. In our problem, recursion is used within the `mystery` function to decrementally evaluate values until the base case is reached. Starting from any integer, the function progressively calls itself with `number + 1` until `number` equals zero. Understanding recursion involves recognizing how each call builds upon the previous one, and how the backtracking of results after hitting the base case leads to the final solution.