Final answer:
To find the root using Fixed Point Iteration, function f(x) must be rearranged as x = g(x), and then iterative methods should be applied until the desired tolerance is met. The provided details in the question do not offer a clear rearrangement, so it is advised to find a suitable g(x) function like g(x) = √(4e^{-x} + 3) to proceed with the iteration.
Step-by-step explanation:
To find the root of the function f(x) = x² - 4e^{-x} - 3 on the interval [1, 2] using the Fixed Point Iteration method, we need to rearrange the function to the form x = g(x) and then use iterative techniques to converge on the solution. However, the information provided in the student's question does not directly lead us to a suitable rearrangement for Fixed Point Iteration. Typically, we look for a g(x) such that for any value x in the interval, g(x) is also in the interval (ensuring convergence), and where g'(x) is less than 1 in absolute value to satisfy the error tolerance |f(x)| ≤ 10⁵. Neither the quadratic equation nor other provided formulas appear to be immediately relevant to this task.
Nevertheless, it is possible to provide a general strategy to approach this problem. First, one would have to find a suitable g(x). For example, one might choose g(x) = √(4e^{-x} + 3), as this naturally arises from the given function when we isolate x on one side. Then, start with an initial guess x₀ between 1 and 2 and compute subsequent iterations by evaluating g(x) at the current approximation of the root. Iterate until the absolute value of f(x) is within the specified tolerance of 10⁵.