f(x^2 - 2) &= 3(x^4 - 4x^2 + 4) - 5 \\

["# Understanding and Simplifying the Expression: ( f(x^2 - 2) = 3(x^4 - 4x^2 + 4) - 5 )", "When dealing with mathematical functions, substituting expressions into a given formula can simplify analysis and solve equations more efficiently. In this article, we explore how to understand and simplify the functional expression:", "[\nf(x^2 - 2) = 3(x^4 - 4x^2 + 4) - 5\n]", "## What Is the Function ( f )?", "The given equation defines a functional form:", "[\nf(x^2 - 2) = 3(x^4 - 4x^2 + 4) - 5\n]", "Our goal is to express ( f(u) ) in terms of ( u ), where ( u = x^2 - 2 ). This allows us to analyze the behavior of ( f ), find explicit formulas, and solve related equations.", "### Step 1: Simplify the Right-Hand Side", "Start by simplifying the expression on the right:", "1. Begin with:\n [\n f(x^2 - 2) = 3(x^4 - 4x^2 + 4) - 5\n ]", "2. Expand the multiplication:\n [\n f(x^2 - 2) = 3x^4 - 12x^2 + 12 - 5\n ]", "3. Combine like terms:\n [\n f(x^2 - 2) = 3x^4 - 12x^2 + 7\n ]", "### Step 2: Express Everything in Terms of ( u = x^2 - 2 )", "Since ( f ) is defined in terms of ( x^2 - 2 ), we express ( x^4 ) and ( x^2 ) using ( u ).", "Let\n[\nu = x^2 - 2 \quad \Rightarrow \quad x^2 = u + 2\n]", "Then,\n[\nx^4 = (x^2)^2 = (u + 2)^2 = u^2 + 4u + 4\n]", "### Step 3: Substitute into the Simplified Expression", "Now substitute into the simplified ( f(x^2 - 2) = 3x^4 - 12x^2 + 7 ):", "[\nf(u) = 3(u^2 + 4u + 4) - 12(u + 2) + 7\n]", "### Step 4: Expand and Combine Like Terms", "Expand each term:", "[\nf(u) = 3u^2 + 12u + 12 - 12u - 24 + 7\n]", "Combine like terms:", "- ( 3u^2 )\n- ( 12u - 12u = 0 )\n- ( 12 - 24 + 7 = -5 )", "Thus,", "[\nf(u) = 3u^2 - 5\n]", "## Final Expression for ( f )", "The simplified function is:", "[\nf(u) = 3u^2 - 5\n]", "This holds for all ( u ) such that ( u = x^2 - 2 ), meaning the domain corresponds to values ( u \geq -2 ), since ( x^2 \geq 0 \Rightarrow x^2 - 2 \geq -2 ).", "## Applications: Evaluating ( f ) at ( x^2 - 2 )", "Using ( f(u) = 3u^2 - 5 ), we can compute:", "[\nf(x^2 - 2) = 3(x^2 - 2)^2 - 5\n]", "which matches the original expression upon expansion, confirming correctness.", "### Solving Equations Involving ( f(x^2 - 2) )", "Suppose we want to solve:", "[\nf(x^2 - 2) = 0\n]", "Substitute:", "[\n3(x^2 - 2)^2 - 5 = 0\n]", "Solving:", "1. Add 5 to both sides:\n [\n 3(x^2 - 2)^2 = 5\n ]", "2. Divide by 3:\n [\n (x^2 - 2)^2 = \frac{5}{3}\n ]", "3. Take square roots:\n [\n x^2 - 2 = \pm \sqrt{\frac{5}{3}}\n ]", "4. Solve for ( x^2 ):\n [\n x^2 = 2 \pm \sqrt{\frac{5}{3}}\n ]", "5. Take square roots to find ( x ):\n [\n x = \pm \sqrt{2 \pm \sqrt{\frac{5}{3}}}\n ]", "This provides all real solutions.", "## Summary", "- The given expression simplifies using substitution:\n [\n f(x^2 - 2) = 3(x^4 - 4x^2 + 4) - 5 = 3x^4 - 12x^2 + 7\n ]\n- Expressing ( x^2 ) in terms of ( u = x^2 - 2 ) leads to:\n [\n f(u) = 3u^2 - 5\n ]\n- The function ( f(u) = 3u^2 - 5 ) is defined for ( u \geq -2 ).\n- This form enables efficient evaluation and solving of equations involving ( f(x^2 - 2) ).", "## Why This Matters", "Understanding function substitution and simplification helps:", "- Streamline complex expressions for integration, differentiation, or equation solving.\n- Reveal symmetry and structure in mathematical models.\n- Facilitate operations like finding ranges, inverses, and composites of functions.", "Whether in calculus, algebra, or applied mathematics, mastering such transformations is key to advanced problem solving.", "---", "Keywords: function substitution, f(x² - 2), simplifying expressions, f(u) = 3u² - 5, algebraic simplification, solving equations, domain analysis"]