Round #1037

Given f(x) = x + 1 and g(x) = x^2, is f(g(x)) the same as g(f(x))? Show both and explain.

```csharp using System; class Program { static void Main() { // Define the functions Func<double, double> f = x => x + 1; Func<double, double> g = x => Math.Pow(x, 2); // Calculate f(g(x)) and g(f(x)) double x = 2; // example value double fg_x = f(g(x)); double gf_x = g(f(x)); Console.WriteLine("f(g(x)) = " + fg_x); Console.WriteLine("g(f(x)) = " + gf_x); } } ```

To determine if f(g(x)) is the same as g(f(x)), we need to calculate both compositions. f(g(x)) = f(x^2) = x^2 + 1 g(f(x)) = g(x + 1) = (x + 1)^2 = x^2 + 2x + 1 Since x^2 + 1 is not equal to x^2 + 2x + 1, f(g(x)) is not the same as g(f(x)). Here is a simple C# code snippet to demonstrate this: ```csharp using System; class Program { static void Main() { // Define a function f(x) = x + 1 Func<double, double> f = x => x + 1; // Define a function g(x) = x^2 Func<double, double> g = x => Math.Pow(x, 2); // Test with a value, for example x = 2 double x = 2; double fgx = f(g(x)); double gfx = g(f(x)); Console.WriteLine($"f(g({x})) = {fgx}"); Console.WriteLine($"g(f({x})) = {gfx}"); } } ``` This code will output the values of f(g(x)) and g(f(x)) for a given x, demonstrating that they are not equal. The explanation of the difference requires more text to fully understand the concept of function composition.