#### 49.97Question: Let $ f(x^2 + 2) = x^4 + 4x^2 + 3 $. Find $ f(x^2 - 2) $.
["Understanding and Finding $ f(x^2 - 2) $ Given $ f(x^2 + 2) = x^4 + 4x^2 + 3 $", "Understanding how to manipulate functions defined on transformed inputs is a crucial skill in algebra and functional equations. In this article, we explore how to determine $ f(x^2 - 2) $ based on the given expression:\n$$\nf(x^2 + 2) = x^4 + 4x^2 + 3.\n$$", "---", "### Step 1: Analyze the structure of the given function", "We are told:\n$$\nf(x^2 + 2) = x^4 + 4x^2 + 3.\n$$\nObserve that the left-hand side shows $ f $ evaluated at $ x^2 + 2 $, while the right-hand side is a polynomial in $ x $. To find $ f(u) $ for a general input $ u $, we want to express the right-hand side purely in terms of $ u $.", "Let:\n$$\nu = x^2 + 2 \quad \Rightarrow \quad x^2 = u - 2.\n$$\nNow express $ x^4 $ in terms of $ u $:\n$$\nx^4 = (x^2)^2 = (u - 2)^2 = u^2 - 4u + 4.\n$$", "Now substitute into the original equation:\n$$\nf(u) = x^4 + 4x^2 + 3 = (u^2 - 4u + 4) + 4(u - 2) + 3.\n$$", "---", "### Step 2: Simplify the expression", "Expand and combine terms:\n$$\nf(u) = u^2 - 4u + 4 + 4u - 8 + 3 = u^2 + ( -4u + 4u ) + (4 - 8 + 3).\n$$\n$$\nf(u) = u^2 - 1.\n$$", "So, we deduce:\n$$\nf(u) = u^2 - 1.\n$$", "---", "### Step 3: Evaluate $ f(x^2 - 2) $", "Now that we know $ f(u) = u^2 - 1 $, substitute $ u = x^2 - 2 $:\n$$\nf(x^2 - 2) = (x^2 - 2)^2 - 1.\n$$", "Expand:\n$$\n(x^2 - 2)^2 = x^4 - 4x^2 + 4,\n$$\nso:\n$$\nf(x^2 - 2) = x^4 - 4x^2 + 4 - 1 = x^4 - 4x^2 + 3.\n$$", "---", "### Final Answer", "$$\n\boxed{f(x^2 - 2) = x^4 - 4x^2 + 3}\n$$", "This derivation shows how functional substitution and algebraic manipulation enable transformation of complex expressions into elegant closed forms. Mastering such techniques empowers deeper understanding of function behavior and composition.", "---", "Keywords: $ f(x^2 + 2) = x^4 + 4x^2 + 3 $, find $ f(x^2 - 2) $, functional substitution, algebraic transformation, polynomial function, substitution method."]
