2Question: Let $ f $ be a function such that $ f(x^2 + 3) = 2x^4 + 10x^2 + 11 $. Find $ f(x^2 - 3) $.
["Title: How to Find $ f(x^2 - 3) $ Given $ f(x^2 + 3) = 2x^4 + 10x^2 + 11 $: A Step-by-Step Guide", "Meta Description:\nLearn how to determine $ f(x^2 - 3) $ using transformation techniques and substitution, based on the functional equation $ f(x^2 + 3) = 2x^4 + 10x^2 + 11 $. Discover step-by-step reasoning and practical applications.", "---", "## Introduction", "Understanding functional equations is a powerful tool in algebra, allowing us to deduce complex function behaviors from compact expressions. In this article, we explore a specific functional definition:\nSuppose $ f(x^2 + 3) = 2x^4 + 10x^2 + 11 $.\nOur goal is to find $ f(x^2 - 3) $—a transformation that reveals deeper insight into the function $ f $. This process illustrates universal techniques for manipulating functions defined implicitly.", "---", "### Step 1: Let’s Simplify the Given Equation", "We are given:\n$$\nf(x^2 + 3) = 2x^4 + 10x^2 + 11\n$$", "Observe that the right-hand side is a quartic in $ x $, expressible entirely in terms of $ x^2 $. This suggests a substitution involving $ u = x^2 $ will streamline the analysis.", "Let $ u = x^2 $. Then $ x^4 = u^2 $, and rewrite the equation:\n$$\nf(u + 3) = 2u^2 + 10u + 11\n$$", "---", "### Step 2: Express $ f(t) $ in Terms of $ t $", "We now aim to find a closed-form expression for $ f(t) $. Since $ f(u + 3) = 2u^2 + 10u + 11 $, define a new variable:\nLet $ t = u + 3 \Rightarrow u = t - 3 $", "Substitute into the expression:\n$$\nf(t) = 2(t - 3)^2 + 10(t - 3) + 11\n$$", "Now expand each term:\n- $ (t - 3)^2 = t^2 - 6t + 9 $\n- $ 2(t^2 - 6t + 9) = 2t^2 - 12t + 18 $\n- $ 10(t - 3) = 10t - 30 $\n- Add constant: $ +11 $", "Now combine all terms:\n$$\nf(t) = 2t^2 - 12t + 18 + 10t - 30 + 11\n$$\n$$\nf(t) = 2t^2 - 2t - 1\n$$", "Thus, the function is:\n$$\nf(t) = 2t^2 - 2t - 1\n$$", "---", "### Step 3: Evaluate $ f(x^2 - 3) $", "Now compute $ f(x^2 - 3) $ using the derived expression:\n$$\nf(x^2 - 3) = 2(x^2 - 3)^2 - 2(x^2 - 3) - 1\n$$", "Expand each term:\n- $ (x^2 - 3)^2 = x^4 - 6x^2 + 9 $\n- $ 2(x^4 - 6x^2 + 9) = 2x^4 - 12x^2 + 18 $\n- $ -2(x^2 - 3) = -2x^2 + 6 $\n- Constant: $ -1 $", "Combine everything:\n$$\nf(x^2 - 3) = 2x^4 - 12x^2 + 18 - 2x^2 + 6 - 1\n$$\n$$\n= 2x^4 - 14x^2 + 23\n$$", "---", "### Final Answer", "$$\n\boxed{f(x^2 - 3) = 2x^4 - 14x^2 + 23}\n$$", "This derivation demonstrates a powerful method: by substituting $ u = x^2 $ and changing the functional variable, we transformed a hidden algebraic structure into a standard polynomial form. From this base, evaluating at shifted inputs becomes straightforward.", "---", "### Why This Matters", "Functional equations like these appear in advanced algebra, optimization problems, and even machine learning models where unknown functional forms are inferred from samples. Mastering such techniques empowers students and researchers to decode complex relationships efficiently.", "---", "Keywords: functional equation, $ f(x^2 + 3) = 2x^4 + 10x^2 + 11 $, $ f(x^2 - 3) $, substitution method, algebraic manipulation, function transformation, polynomial derivation", "Related Topics: solving functional equations, polynomial function reconstruction, variable substitution in algebra, finding inverse functions.", "---", "Have you tackled similar functional expressions? Let us know in the comments! Stay sharp, keep exploring functions."]
