Question: Given $ f(2x + 1) = 4x^2 + 12x + 9 $, find $ f(x^2 - 3) $.
["Title: How to Find $ f(x^2 - 3) $ Given $ f(2x + 1) = 4x^2 + 12x + 9 $ – A Step-by-Step Guide", "Meta Description:\nStruggling to find $ f(x^2 - 3) $ when $ f(2x + 1) = 4x^2 + 12x + 9 $? Learn how to transform the function step-by-step with a detailed example and key insights.", "---", "### Introduction", "Functional equations are powerful tools in algebra that help us uncover hidden forms of functions. One common challenge students face is finding $ f(x^2 - 3) $ when the function is defined in terms of a linear expression like $ f(2x + 1) $. In this article, we’ll walk through the process of determining $ f(x^2 - 3) $ from the given definition:\n$$\nf(2x + 1) = 4x^2 + 12x + 9\n$$", "Understanding this process not only sharpens your algebraic skills but also prepares you for advanced problem-solving in calculus and higher mathematics.", "---", "### Step 1: Rewrite the Function in Terms of a New Variable", "We begin by letting\n$$\nu = 2x + 1\n$$\nOur goal is to express $ f(u) $ in terms of $ u $, so we solve for $ x $ in terms of $ u $:\n$$\nu = 2x + 1 \Rightarrow x = \frac{u - 1}{2}\n$$", "---", "### Step 2: Substitute into the Given Expression", "Now, substitute $ x = \frac{u - 1}{2} $ into the right-hand side:\n$$\nf(u) = 4\left( \frac{u - 1}{2} \right)^2 + 12\left( \frac{u - 1}{2} \right) + 9\n$$", "Simplify each term:\n- First term:\n$$\n4 \cdot \left( \frac{u - 1}{2} \right)^2 = 4 \cdot \frac{(u - 1)^2}{4} = (u - 1)^2\n$$\n- Second term:\n$$\n12 \cdot \frac{u - 1}{2} = 6(u - 1)\n$$\n- Third term remains $ +9 $", "So:\n$$\nf(u) = (u - 1)^2 + 6(u - 1) + 9\n$$", "Expand and simplify:\n$$\n(u - 1)^2 = u^2 - 2u + 1\n$$\n$$\n6(u - 1) = 6u - 6\n$$\nAdd all terms:\n$$\nf(u) = u^2 - 2u + 1 + 6u - 6 + 9 = u^2 + 4u + 4\n$$", "Thus,\n$$\nf(u) = u^2 + 4u + 4\n$$", "---", "### Step 3: Express $ f $ in Terms of Any Input", "Since $ u $ is a placeholder, we write:\n$$\nf(x) = x^2 + 4x + 4\n$$", "---", "### Step 4: Evaluate $ f(x^2 - 3) $", "Now substitute $ x^2 - 3 $ into $ f $:\n$$\nf(x^2 - 3) = (x^2 - 3)^2 + 4(x^2 - 3) + 4\n$$", "Expand each term:\n- $ (x^2 - 3)^2 = x^4 - 6x^2 + 9 $\n- $ 4(x^2 - 3) = 4x^2 - 12 $\n- Constant: $ +4 $", "Add them together:\n$$\nf(x^2 - 3) = x^4 - 6x^2 + 9 + 4x^2 - 12 + 4 = x^4 - 2x^2 + 1\n$$", "---", "### Final Answer", "$$\nf(x^2 - 3) = x^4 - 2x^2 + 1\n$$", "---", "### Key Takeaways", "- Use substitution to transform functional equations into polynomial forms.\n- Express $ f(u) $ explicitly by replacing the input variable.\n- Replacing the function input with $ x^2 - 3 $ allows evaluation for any expression.", "This method is broadly applicable for solving functional equations involving polynomial functions. Whether you're preparing for exams or solving real-world modeling problems, mastering such substitutions is crucial.", "---", "Want more? Explore how to differentiate functional forms or find inverse functions based on given mappings. Keep practicing and confidently tackle functional equations!", "---", "Keywords:\n$ f(2x + 1) = 4x^2 + 12x + 9 $, find $ f(x^2 - 3) $, functional equations, how to find $ f(x) $, substitution method, algebra tutorial, mathematical problem-solving.", "Related searches:\nhow do you find $ f(x) $ from $ f(2x+1) $, step-by-step function transformation, evaluate composite functions from given expressions."]
