Question

Consider the following function: int func(int x) { if (x == 0) return 2; else if (x == 1) return 3; else return (func(x - 1) + func(x - 2)); } What is the output of the following statements? a. cout << func(0) << endl; b. cout << func(1) << endl; c. cout << func(2) << endl; d. cout << func(5) << endl;

Short answer

a. 2 b. 3 c. 5 d. 21

Step-by-step solution

Understanding the Function

The given function is a recursive function. It returns 2 if the input is 0, and 3 if the input is 1. For any other input, it computes the sum of the function applied to the two preceding numbers: \( func(x) = func(x-1) + func(x-2) \).

Calculate func(0)

For statement \(a\), when the input to the function is 0:- According to the function definition, \( func(0) = 2 \).The output will be 2.

Calculate func(1)

For statement \(b\), when the input to the function is 1:- The function returns 3 based on the direct condition for \( func(1) \).The output will be 3.

Calculate func(2)

For statement \(c\), when the input to the function is 2:- The function uses recursion: \( func(2) = func(1) + func(0) \).- Substitute the values calculated previously: \( func(2) = 3 + 2 = 5 \).The output will be 5.

Calculate func(5)

For statement \(d\), when the input to the function is 5, we use multiple recursion calls:- \( func(5) = func(4) + func(3) \)- To find \( func(4) = func(3) + func(2) \) and \( func(3) = func(2) + func(1) \)- Calculate these values further: - Already calculated \( func(2) = 5 \), \( func(1) = 3 \), and \( func(0) = 2 \). - \( func(3) = 5 + 3 = 8 \) - \( func(4) = 8 + 5 = 13 \) - \( func(5) = 13 + 8 = 21 \)The output will be 21.

Key concepts

Recursion

Recursion is a fundamental programming concept where a function calls itself in order to solve a problem. This technique can simplify problems into smaller, more manageable ones by repeatedly breaking them down into sub-problems of the same type. The given function `func` in the exercise is an example of a recursive function. It repeatedly calls itself with decremented input values based on the conditions defined within the function body.

To understand recursion, it's crucial to visualize the call stack, where each function call is stacked until a condition stops further calls. This stopping condition is known as the base case. Recursion is particularly useful for problems that exhibit optimal substructure and can be defined in terms of themselves, such as the classic Fibonacci sequence.

Function Calls

Function calls are the mechanisms that allow functions to be executed within a program. When a function is called, control is passed to that function code block, allowing the operations defined there to be executed with any supplied arguments.

In the context of the recursive `func` function, every time it calls itself, a new function call is added to the call stack. This process continues until the function encounters a base case—either `x == 0` or `x == 1`. Each call to a function returns a value that will be used in calculations higher in the stack, until the original function call finally returns the total value to the main program. Remember to ensure that all function calls eventually lead to a base case to prevent infinite recursion.

Base Case

In recursive functions, the base case is the condition that stops the recursion. It acts as a termination point, allowing the function to exit without making any further recursive calls. This prevents the program from entering an infinite loop and potentially crashing due to excessive use of resources.

In the `func` function from the exercise, there are two base cases:

• If `x == 0`, the function returns 2.
• If `x == 1`, the function returns 3.

These base cases ensure that once the input reaches 0 or 1, the recursion stops, and the function begins to return values back through the recursive call stack. It is vital for recursive functions to have a well-defined base case to ensure that they do not run indefinitely.

C++ Programming

C++ Programming provides the capability to write efficient and clear recursive functions, as demonstrated in the given exercise. In C++, recursion is implemented by writing functions that call themselves with modified parameters until a base case is reached. This language supports recursion seamlessly, allowing programmers to tackle complex problems by defining a recursive logic succinctly.

When writing recursive functions in C++, keep these tips in mind:

• Always define a base case to prevent infinite recursion and stack overflow errors.
• Consider using memoization to improve performance if the same results are computed multiple times.
• Be mindful of memory and performance issues, as recursion can be costly in terms of resources for deep recursive calls.

C++'s Function Call stack is utilized efficiently in recursive programming, handling each call's data until the recursion unwinds. This showcases C++'s strength in supporting advanced programming techniques like recursion.