Evaluating composed functions involves understanding how to combine two functions and then determine their output based on a specific input. When given two functions, such as f(x) = x² and g(x) = x - 1, the composition f(g(x)) can be found by substituting g(x) into f(x). This process begins by calculating f(g(x)) = f(x - 1), which simplifies to (x - 1)². By applying the FOIL method, we expand this to x² - 2x + 1, providing the composed function.

To evaluate this composition at a specific value, such as 3, we replace x in the composed function with 3. This results in f(g(3)) = 3² - 2(3) + 1, which simplifies to 9 - 6 + 1 = 4. Thus, f(g(3)) = 4.

Alternatively, a shortcut method can be employed for evaluating composed functions. This involves first calculating the inner function at the desired input. For instance, finding g(3) gives us 3 - 1 = 2. We then substitute this result into the outer function: f(g(3)) = f(2), which simplifies to 2² = 4. This method can be quicker but is not always applicable, especially when the problem requires finding the composition before evaluating.

In summary, while both methods yield the same result, the choice of method may depend on the specific requirements of the problem. Understanding how to compose functions and evaluate them effectively is crucial in mastering function operations.