5Question: In complex analysis, if $ f(z) = z^2 $ and $ g(z) = \text{Re}(z) + i\text{Im}(z) $, compute $ f(g(3 + 4i)) $ in the form $ a + bi $.

["Title: Compute $ f(g(3 + 4i)) $: Step-by-Step in Complex Analysis", "---", "Introduction", "In complex analysis, evaluating functions of complex inputs often involves carefully applying definitions and operations. This article walks through computing $ f(g(3 + 4i)) $, where $ f(z) = z^2 $ and $ g(z) = \ ext{Re}(z) + i\ ext{Im}(z) $, resulting in the final expression in the standard form $ a + bi $.", "---", "Step-by-Step Computation", "1. Understand $ g(z) $\n The function $ g(z) $ extracts the real and imaginary parts of a complex number $ z $, then returns them as $ g(z) = \ ext{Re}(z) + i\ ext{Im}(z) $.\n This is simply a restatement of $ z $ itself in complex form — $ g(z) = z $.\n Therefore,\n $$\n g(3 + 4i) = 3 + 4i\n $$", "2. Apply $ f(z) = z^2 $\n Now substitute $ z = g(3 + 4i) = 3 + 4i $ into $ f(z) $:\n $$\n f(g(3 + 4i)) = f(3 + 4i) = (3 + 4i)^2\n $$", "3. Square the complex number\n Expand $ (3 + 4i)^2 $:\n $$\n (3 + 4i)^2 = 3^2 + 2 \cdot 3 \cdot 4i + (4i)^2 = 9 + 24i + 16i^2\n $$\n Since $ i^2 = -1 $, we substitute:\n $$\n 9 + 24i + 16(-1) = 9 + 24i - 16 = -7 + 24i\n $$", "---", "Final Answer", "Thus,\n$$\n\boxed{f(g(3 + 4i)) = -7 + 24i}\n$$", "---", "Conclusion", "This problem illustrates the importance of clearly interpreting functions in complex analysis. Recognizing that $ g(z) = z $ simplifies evaluation significantly. Squaring $ 3 + 4i $ yields $ -7 + 24i $, demonstrating how algebraic operations combine real and imaginary components through squaring. Understanding these foundational steps builds confidence for more advanced work in complex function theory."]