Solution: First, compute $ g(3 + 4i) = 3 + 4i $, since $ g(z) $ returns $ z $ itself. Then, $ f(g(3 + 4i)) = f(3 + 4i) = (3 + 4i)^2 = 9 + 24i + 16i^2 = 9 + 24i - 16 = -7 + 24i $. The result is $\boxed{-7 + 24i}$.
["Title: Understanding Complex Function Composition: Evaluating $ f(g(3 + 4i)) $", "Meta Description:\nExplore the step-by-step evaluation of complex function composition $ f(g(3 + 4i)) $, where $ g(z) = z $ and $ f(z) = z^2 $. Learn how complex numbers are squared and simplified using fundamental algebraic rules.", "---", "## Mastering Complex Function Composition: A Step-by-Step Guide to $ f(g(3 + 4i)) $", "In complex analysis, evaluating functions composed with complex numbers is a fundamental skill. Today, we’ll explore a clear, educational example: computing $ f(g(3 + 4i)) $, where two functions act on the complex number $ z = 3 + 4i $. This example illustrates how to compute function composition with real and imaginary components, leveraging basic algebra and imaginary unit properties.", "### Problem Statement", "We are given:", "- $ g(z) = z $, meaning $ g $ returns the input unchanged.\n- $ f(z) = z^2 $, so $ f(g(3 + 4i)) = f(3 + 4i) $.", "We must compute $ f(g(3 + 4i)) $ and arrive at the final result:\n[\n\boxed{-7 + 24i}\n]", "---", "### Step 1: Compute $ g(3 + 4i) $", "Since $ g(z) $ returns the input itself:\n[\ng(3 + 4i) = 3 + 4i\n]", "So, the output of $ g $ becomes the complex number $ 3 + 4i $, which now serves as the input to $ f $.", "---", "### Step 2: Compute $ f(3 + 4i) $", "The function $ f $ computes the square of its input. For complex numbers, squaring involves expanding $ (a + bi)^2 $ using the identity $ i^2 = -1 $.", "Let’s compute:\n[\nf(3 + 4i) = (3 + 4i)^2\n]", "Apply the formula $ (a + b)^2 = a^2 + 2ab + b^2 $:\n[\n(3 + 4i)^2 = 3^2 + 2 \cdot 3 \cdot 4i + (4i)^2\n]", "Calculate each term:\n- $ 3^2 = 9 $\n- $ 2 \cdot 3 \cdot 4i = 24i $\n- $ (4i)^2 = 16i^2 = 16(-1) = -16 $", "Add the results:\n[\n9 + 24i - 16 = (9 - 16) + 24i = -7 + 24i\n]", "---", "### Final Result:\n[\nf(g(3 + 4i)) = \boxed{-7 + 24i}\n]", "---", "### Conclusion", "This example demonstrates that when a function $ g $ returns its input unchanged, composing it with $ f(z) = z^2 $ results in simply squaring the complex number $ 3 + 4i $. Understanding such compositions helps build a solid foundation for working with functions in complex analysis, algebra, and applied mathematics. The detailed breakdown ensures clarity, making it easier to grasp even complex-looking expressions like squaring $ 3 + 4i $.", "Whether you're a student, educator, or professional in STEM fields, mastering these steps enhances your ability to tackle complex number computations with confidence.", "---", "### Key Takeaways:", "- $ g(z) = z $ acts as the identity function.\n- $ f(z) = z^2 $ applies squaring, extended naturally to complex numbers.\n- Rule: $ i^2 = -1 $ is essential for simplifying squares.\n- Step-by-step evaluation ensures accuracy and reinforces conceptual understanding.", "---", "Optimize your learning with clear examples—complex functions don’t have to be intimidating when broken down clearly!"]
