Define the function f(x)=2x/1−x^2. Find the number of distinct real solutions of the equation f^(5)(x)=x.
Details and assumptions
f^(n)(x) denotes the function f applied n times. In particular, f^(5)(x)=f(f(f(f(f(x))))).
To find the number of distinct real solutions of the equation f^(5)(x) = x, we first need to compute the fifth composition of the function f(x) = 2x/(1−x^2).
Let's start by finding the first composition of f(x). We substitute the function f(x) into itself:
f(f(x)) = f(2x/(1−x^2)) = 2(2x/(1−x^2)) / (1−(2x/(1−x^2)))^2
Next, we find the composition f(f(f(x))) by substituting f(f(x)) into f(x):
f(f(f(x))) = f(2(2x/(1−x^2)) / (1−(2x/(1−x^2)))^2) = 2(2(2x/(1−x^2)) / (1−(2x/(1−x^2)))^2) / (1−(2(2x/(1−x^2)) / (1−(2x/(1−x^2)))^2))^2
We continue this process until we find f^(5)(x):
f(f(f(f(f(x))))) = 2(2(2(2(2x/(1−x^2)) / (1−(2x/(1−x^2)))^2) / (1−(2(2x/(1−x^2)) / (1−(2x/(1−x^2)))^2))^2) / (1−(2(2(2x/(1−x^2)) / (1−(2x/(1−x^2)))^2) / (1−(2(2x/(1−x^2)) / (1−(2x/(1−x^2)))^2))^2))^2
Now, we need to solve the equation f^(5)(x) = x. We set the above expression equal to x and simplify the equation:
2(2(2(2(2x/(1−x^2)) / (1−(2x/(1−x^2)))^2) / (1−(2(2x/(1−x^2)) / (1−(2x/(1−x^2)))^2))^2) / (1−(2(2(2x/(1−x^2)) / (1−(2x/(1−x^2)))^2) / (1−(2(2x/(1−x^2)) / (1−(2x/(1−x^2)))^2))^2))^2 = x
To find the number of distinct real solutions, we need to solve this equation numerically using a graphing calculator, computational software, or by graphing and analyzing the equation visually. By examining the graph or using numerical approximation methods, we can determine the number of distinct real solutions to the equation f^(5)(x) = x.
Note: Due to the complexity of the equation, the numerical solution might not be easily obtainable without the aid of software or tools.