{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).​​ 


b. Consider the polynomial g(x) = f(f(x)). Write g in terms of a and powers of x. What is its degree?

Verified step by step guidance

1. First, understand that a fixed point of a function f(x) is a value x such that f(x) = x. For the function f(x) = ax(1 - x), we need to find f(f(x)) to form the polynomial g(x).
2. Start by substituting f(x) into itself: f(f(x)) = a(f(x))(1 - f(x)). This means we need to substitute f(x) = ax(1 - x) into the expression again.
3. Calculate f(x) = ax(1 - x) and substitute it into the expression: f(f(x)) = a(ax(1 - x))(1 - ax(1 - x)).
4. Simplify the expression: Expand the terms inside the parentheses and multiply them out. This will involve expanding (1 - ax(1 - x)) and multiplying it by ax(1 - x).
5. Determine the degree of the polynomial g(x): After simplifying, observe the highest power of x in the expression. The degree of g(x) will be the highest power of x present in the expanded form.

Key Concepts

Fixed Points

A fixed point of a function is a value of x for which the function evaluates to the same value, meaning f(x) = x. This concept is crucial in understanding the behavior of functions, particularly in iterative processes and stability analysis. In the context of the given function f(x) = ax(1 - x), finding fixed points involves solving the equation ax(1 - x) = x.

Composition of Functions

The composition of functions involves applying one function to the result of another. In this case, g(x) = f(f(x)) means we first apply f to x, and then apply f again to the result. Understanding function composition is essential for manipulating and simplifying expressions, especially when dealing with polynomials and their degrees.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It provides insight into the polynomial's behavior, such as the number of roots and the end behavior of the graph. For the polynomial g(x) derived from f(f(x)), determining its degree involves analyzing the composition and identifying the maximum exponent of x in the resulting expression.