Question

Consider the following function: int test(int x, int y) { if (x == y) return x; else if (x > y) return (x + y); else return test(x + 1, y - 1); } What is the output of the following statements? a. cout << test(5, 10) << endl; b. cout << test(3, 9) << endl;

Short answer

a. 15; b. 6

Step-by-step solution

Understand the Function Structure

The function `test` takes two integers, `x` and `y`, as inputs and compares them. It has three scenarios: 1. If `x` equals `y`, it returns `x`. 2. If `x` is greater than `y`, it returns the sum of `x` and `y`. 3. If `x` is less than `y`, it recursively calls itself with `x + 1` and `y - 1`.

Analyzing test(5, 10)

For `test(5, 10)`, initialize with `x = 5` and `y = 10`. Since `x < y`, proceed to the recursive call: `test(5+1, 10-1)` or `test(6, 9)`. Repeat this until `x == y`. - `test(6, 9)`: x < y, call `test(7, 8)`. - `test(7, 8)`: x < y, call `test(8, 7)`. - `test(8, 7)`: x > y, return `x + y = 15`.

Find the Output for (a) cout

Each recursive step modifies `x` and `y` with `x + 1` and `y - 1`, until `test(8, 7)` where x > y, triggering `x + y = 15`. The output for `cout << test(5, 10)` is `15`.

Analyzing test(3, 9)

For `test(3, 9)`, begin with `x = 3` and `y = 9`. Recursive calls continue until `x == y`. - `test(3, 9)`: x < y, call `test(4, 8)`. - `test(4, 8)`: x < y, call `test(5, 7)`. - `test(5, 7)`: x < y, call `test(6, 6)`. - `test(6, 6)`: x == y, return `x = 6`.

Find the Output for (b) cout

Each recursive step converges towards making `x` equal to `y`. At `test(6, 6)`, a return of `6` is triggered. Thus, the output for `cout << test(3, 9)` is `6`.

Key concepts

Understanding Recursive Functions

A recursive function in C++ is a function that calls itself in order to solve a problem. This is done by breaking down a problem into smaller, more manageable sub-problems of the same type. In the exercise provided, the `test` function is a clear example of recursion in action. It includes a base case to stop the recursion and prevent it from continuing indefinitely. The base case in this function occurs when the if-statement condition `x == y` is satisfied. At this point, instead of making further recursive calls, the function simply returns a value, thus stopping the recursion. **Key points of recursive functions:**

• They must have a clear base case that stops the recursion.
• The problem should be broken into smaller sub-problems.
• Careful management is needed to prevent infinite recursion.

The effect of this recursive approach is the repetitive adjustment of `x` and `y` through `x + 1` and `y - 1`, moving `x` and `y` towards each other until one of the base conditions is met.

Role of Conditional Statements

Conditional statements, such as if-else, are crucial in controlling the flow of recursive functions. In the `test` function, these statements decide which pathway the program takes, based on the relationship between `x` and `y`. **Types of conditions used in the function:**

• **Equality check (`x == y`)**: When both integers are equal, recursion stops and `x` is returned. This acts as a safeguard, preventing further unnecessary calculations.
• **Greater than check (`x > y`)**: If `x` surpasses `y`, the condition leads to the sum of `x` and `y` being returned. This shows the completion of a certain logical pathway within the recursive sequence.
• **Less than check (else)**: If neither of the above conditions is met, the function proceeds with the recursive call. It adjusts the values to progress towards the base case.

By leveraging conditional statements, programmers can ensure that functions react differently to varying inputs, enabling complex decision-making processes within recursive functions.

Performing Function Analysis

Function analysis involves understanding what a function does step-by-step to predict its output for different inputs. For the provided `test` function, the analysis reveals its behavior through recursive and conditional logic. In analyzing `test(5, 10)`, we saw:

• The recursive progression, with initial `x = 5` and `y = 10`, adjusting through recursive calls.
• The sequence follows: `test(6, 9)`, `test(7, 8)`, and finally, `test(8, 7)`, where it breaks the recursion and outputs `15` (as `8 + 7`).

When examining `test(3, 9)`, we found:

• The process starts with `x = 3` and `y = 9`.
• It proceeds until the equal state `test(6, 6)` is reached, resulting in the return of `6`.

Such analyses involve a careful tracing of the recursive pathways and understanding the change in values at each step. By doing this, one can not only predict outputs but also improve or debug the function efficiently.