Question

Consider the following recursive function: int mystery(int number) //Line 1 { if (number == 0) //Line 2 return number; //Line 3 else //Line 4 return(number + mystery(number – 1)); //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(5) 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: when number == 0. b) General case: return(number + mystery(number - 1)). c) Valid: non-negative integers. d) Value: 0. e) Value: 15. f) Invalid for -3; leads to infinite recursion.

Step-by-step solution

Identify the Base Case

The base case is the condition in a recursive function that stops the recursion. In the given function, the base case is when \( \text{number} = 0 \). When this condition is true, the function returns \( \text{number} \). This is indicated by Line 2 and Line 3 in the code.

Identify the General Case

The general case is what occurs if the base case condition is not met. For this function, the general case is represented by \( \text{return(number + mystery(number - 1))} \). This loop continues the recursion by reducing the value of 'number' by 1 at each call until the base case is hit.

Determine Valid Values for Parameters

To determine valid inputs for the function, consider the stopping condition (base case) and input constraints. Here, valid inputs are non-negative integers (0, 1, 2,...) because recursion stops safely at 0. Negative integers would cause infinite recursion and are thus invalid.

Calculate mystery(0)

Since 0 is the base case, calling \( \text{mystery(0)} \) satisfies the base case condition. Therefore, the function returns 0.

Calculate mystery(5)

Calculate the result by the recursive calls:1. \( \text{mystery(5)} = 5 + \text{mystery(4)} \)2. \( \text{mystery(4)} = 4 + \text{mystery(3)} \)3. \( \text{mystery(3)} = 3 + \text{mystery(2)} \)4. \( \text{mystery(2)} = 2 + \text{mystery(1)} \)5. \( \text{mystery(1)} = 1 + \text{mystery(0)} \)6. \( \text{mystery(0)} = 0 \) (base case)Each step adds up to give the sum: \( 5 + 4 + 3 + 2 + 1 + 0 = 15 \). So, \( \text{mystery(5)} = 15 \).

Analyze mystery(-3)

Calling \( \text{mystery(-3)} \) is not valid because the function only reliably terminates for non-negative integers. With negative numbers, the base case \( \text{number == 0} \) is never reached, leading to infinite recursion and eventually a stack overflow error.

Key concepts

Base Case

In recursion, the base case is crucial as it defines when the recursive function should stop calling itself. Without a base case, the recursion would continue indefinitely, resulting in what is known as an infinite recursion. In the context of the `mystery` function, the base case is clearly defined in Line 2 of the code: `if (number == 0)`. This means that whenever the function is called with the parameter `number` equal to 0, the recursion stops, and the function returns 0 as specified in Line 3.

Having a well-defined base case ensures that the recursive process terminates safely. It acts as a safeguard to prevent the function from endlessly calling itself. In other words, the base case is like the stopping point in a race; it tells you where you must end to avoid going off track.

When analyzing recursive functions, always check for an appropriate base case to ensure the function behaves as expected. This fundamental concept aids in preventing stack overflow errors.

Recursive Function

A recursive function is a powerful tool in programming where a function calls itself to solve a problem. The recursive function is primarily split into two parts: the base case and the general (or recursive) case. Understanding the general case is pivotal for grasping recursion fully.

In the `mystery` function, once the base case is checked and not satisfied, the general case takes over, defined in Line 5: `return (number + mystery(number - 1));`. This line showcases how the function repeatedly calls itself to work towards the base case, decreasing the `number` by 1 with each call. This process continues recursively, accumulating the sum of numbers down to the base case.

To decode what a recursive function achieves, follow its general case until you reach a clear understanding of the outcome at each recursive step. Recursion is much like solving a puzzle, where each piece needs to fit into the next to see the complete picture. For the `mystery` function, it's like summing a series of numbers from the input down to zero.

Parameter Validity

In recursive functions, parameter validity outlines the range of inputs that the function can handle safely. This verification is essential to avoid runtime errors like infinite recursion. For the `mystery` function, valid inputs are those that include the non-negative integers (0 and positive numbers).

When `mystery` is called with non-negative integers, it will always safely hit the base case and stop ongoing recursion. However, if negative numbers are passed as parameters, the base case condition `number == 0` will never be satisfied. This situation leads to the dangers of infinite recursion, where the function continues calling itself without ever reaching a condition to stop.

Developers must always ensure they understand the parameter limitations of recursive functions and implement guard clauses or checks to handle or reject invalid inputs. This practice not only enhances the robustness of code but also ensures functions do not lead to unintended errors or crashes.