Question: Let $ g(x^2 + 1) = x^4 + 2x^2 + 2 $. Find $ g(x^2 - 1) $.
["Title: How to Find $ g(x^2 - 1) $ Given $ g(x^2 + 1) = x^4 + 2x^2 + 2 $ – A Step-by-Step Solution", "Meta Description:\nLearn how to determine $ g(x^2 - 1) $ when given $ g(x^2 + 1) = x^4 + 2x^2 + 2 $. This step-by-step guide breaks down substitution and function transformation to solve for $ g $ and evaluate at a new input.", "---", "## Introduction", "Functional equations often challenge students and math enthusiasts alike. In this article, we’ll explore a classic problem: given a composite function $ g(x^2 + 1) = x^4 + 2x^2 + 2 $, determine the explicit value of $ g(x^2 - 1) $. By applying strategic substitutions and understanding how to reverse-engineer $ g $, we uncover a powerful method for evaluating complex function forms.", "---", "## Understanding the Given Functional Equation", "We are told:", "$$\ng(x^2 + 1) = x^4 + 2x^2 + 2\n$$", "Our goal is to find $ g(x^2 - 1) $. To do this, we must determine the general form of $ g(u) $, where $ u = x^2 + 1 $, and then substitute $ u = x^2 - 1 $.", "---", "## Step 1: Express the Right-Hand Side in Terms of $ x^2 + 1 $", "Notice that the right-hand side is a polynomial in $ x $, but we aim to write it purely as a function of $ x^2 + 1 $.", "Given:\n$$\ng(x^2 + 1) = x^4 + 2x^2 + 2\n$$", "Let’s express the right-hand side in terms of $ x^2 $. Observe:", "$$\nx^4 + 2x^2 + 2 = (x^2)^2 + 2x^2 + 2\n$$", "Let $ u = x^2 + 1 $. Then $ x^2 = u - 1 $. Substitute into the expression:", "$$\nx^4 + 2x^2 + 2 = (u - 1)^2 + 2(u - 1) + 2\n$$", "Now expand:", "- $ (u - 1)^2 = u^2 - 2u + 1 $\n- $ 2(u - 1) = 2u - 2 $", "So:", "$$\nu^2 - 2u + 1 + 2u - 2 + 2 = u^2 + ( -2u + 2u ) + (1 - 2 + 2) = u^2 + 1\n$$", "Therefore:", "$$\ng(x^2 + 1) = u^2 + 1 = (x^2 + 1)^2 + 1 - x^2 + x^2? \quad \ ext{Wait — actually:} \quad g(u) = u^2 + 1\n$$", "Check: if $ u = x^2 + 1 $, then $ g(u) = u^2 + 1 $. Let’s verify:", "$$\ng(x^2 + 1) = (x^2 + 1)^2 + 1 = x^4 + 2x^2 + 1 + 1 = x^4 + 2x^2 + 2\n$$", "✅ Confirmed. So the function is:", "$$\ng(u) = u^2 + 1\n$$", "---", "## Step 2: Evaluate $ g(x^2 - 1) $", "Now that we know $ g(u) = u^2 + 1 $, substitute $ u = x^2 - 1 $:", "$$\ng(x^2 - 1) = (x^2 - 1)^2 + 1\n$$", "Expand:", "$$\n(x^2 - 1)^2 = x^4 - 2x^2 + 1\n$$", "Add 1:", "$$\nx^4 - 2x^2 + 1 + 1 = x^4 - 2x^2 + 2\n$$", "---", "## Final Answer", "$$\ng(x^2 - 1) = x^4 - 2x^2 + 2\n$$", "---", "## Why This Method Works", "This problem exemplifies how smart substitution simplifies functional equations. Instead of working with abstract substitutions, we:", "- Recognized $ x^2 + 1 $ as a variable $ u $,\n- Expressed the right-hand side purely in terms of $ u $,\n- Derived a closed-form rule for $ g(u) = u^2 + 1 $,\n- Then evaluated at the desired input.", "This technique applies broadly — whether analyzing functional forms in algebra, functional calculus, or algorithm design.", "---", "## Bonus Tip: Practice with Similar Structures", "Try another example:\nIf $ f(2x + 3) = 4x^2 + 12x + 13 $, find $ f(x^2) $.\nLet $ u = 2x + 3 $ → $ x = \frac{u - 3}{2} $, substitute into right-hand side — then express $ f(u) $, then replace $ u $ with $ x^2 $. The method generalizes.", "---", "## Conclusion", "Understanding how to extract and manipulate function rules from composite forms empowers deeper problem-solving skills. Whether in school, exams, or real-world math modeling, knowing how to “unpack” $ g(x^2 + 1) $ gives you full access to $ g(x^2 - 1) $.", "Now you know: $ g(x^2 - 1) = x^4 - 2x^2 + 2 $ — and how to derive it.", "---", "Keywords:\n$ g(x^2 + 1) = x^4 + 2x^2 + 2 $, find $ g(x^2 - 1) $, function substitution, algebra problem solution, functional equations, step-by-step math, how to evaluate function at complex argument."]
