Question: Given the functional equation $ f(x^2 - 3) = 2x^4 - 12x^2 + 13 $, find $ f(x^2 + 1) $.
["# Solving the Functional Equation: Finding $ f(x^2 + 1) $ Given $ f(x^2 - 3) = 2x^4 - 12x^2 + 13 $", "Functional equations appear frequently in advanced algebra and olympiad-style problems, challenging students to uncover hidden patterns and transformations. One such problem involves determining $ f(x^2 + 1) $ when the function $ f $ is defined implicitly through another functional relationship:\n$$\nf(x^2 - 3) = 2x^4 - 12x^2 + 13.\n$$\nIn this article, we’ll walk through the step-by-step method to deduce $ f $ explicitly and then compute $ f(x^2 + 1) $, offering valuable insight into solving functional equations efficiently.", "---", "## Step 1: Understand the Given Functional Equation", "We are given:\n$$\nf(x^2 - 3) = 2x^4 - 12x^2 + 13.\n$$\nOur goal is to find an expression for $ f(y) $, where $ y = x^2 - 3 $, and then substitute $ y = x^2 + 1 $ to compute $ f(x^2 + 1) $.", "---", "## Step 2: Express the Right-Hand Side in Terms of $ y = x^2 - 3 $", "Let $ y = x^2 - 3 $. Then:\n- $ x^2 = y + 3 $\n- $ x^4 = (x^2)^2 = (y + 3)^2 = y^2 + 6y + 9 $", "Now rewrite the right-hand side $ 2x^4 - 12x^2 + 13 $ using these expressions:", "$$\n2x^4 = 2(y^2 + 6y + 9) = 2y^2 + 12y + 18\n$$\n$$\n-12x^2 = -12(y + 3) = -12y - 36\n$$\nAdd all terms:\n$$\n2x^4 - 12x^2 + 13 = (2y^2 + 12y + 18) + (-12y - 36) + 13 = 2y^2 + (12y - 12y) + (18 - 36 + 13)\n$$\n$$\n= 2y^2 - 5\n$$", "Therefore:\n$$\nf(y) = 2y^2 - 5\n$$", "---", "## Step 3: Verify the Expression for $ f(y) $", "To ensure correctness, verify that $ f(x^2 - 3) = 2(x^2 - 3)^2 - 5 $ matches the original:\n$$\n(x^2 - 3)^2 = x^4 - 6x^2 + 9\n\Rightarrow 2(x^4 - 6x^2 + 9) - 5 = 2x^4 - 12x^2 + 18 - 5 = 2x^4 - 12x^2 + 13\n$$\n✓ Matches exactly.", "---", "## Step 4: Compute $ f(x^2 + 1) $", "Now substitute $ y = x^2 + 1 $ into $ f(y) = 2y^2 - 5 $:", "$$\nf(x^2 + 1) = 2(x^2 + 1)^2 - 5\n$$\n$$\n= 2(x^4 + 2x^2 + 1) - 5 = 2x^4 + 4x^2 + 2 - 5 = 2x^4 + 4x^2 - 3\n$$", "---", "## Final Answer", "$$\n\boxed{f(x^2 + 1) = 2x^4 + 4x^2 - 3}\n$$", "---", "## Why This Method Works", "This technique relies on substitution and algebraic manipulation to transform the functional equation into a closed-form expression. By expressing everything in terms of a convenient variable $ y = x^2 - 3 $, we effectually "diagonalize" the functional form, making it easy to evaluate at any transformed input, such as $ x^2 + 1 $.", "---", "## Bonus Insight: Domain Considerations", "Though not required for computation, it's worth noting that for the original equation $ f(x^2 - 3) = 2x^4 - 12x^2 + 13 $, since $ x^2 - 3 $ can take all real values $ \geq -3 $, the function $ f(y) = 2y^2 - 5 $ is valid for $ y \geq -3 $. Since $ x^2 + 1 \geq 1 > -3 $, evaluating at $ x^2 + 1 $ is perfectly valid and lies well within the function’s domain.", "---", "## Conclusion", "Given a functional equation involving compositions like $ f(x^2 - 3) $, substitution and expression in powers of the inner function enable a clean path to determining $ f(y) $. This method—substitution, algebraic simplification, and verification—is a powerful tool in functional equation solving and remains highly applicable in mathematical competition and problem-solving contexts.", "Next time you face $ f(x^2 + a) $, try expressing the right-hand side in terms of $ x^2 - 3 $, then convert everything into powers of that variable—often leads directly to the solution."]
