f(x) = rac{(x+1)^2 + (x-1)^2}{(x-1)(x+1)} - 2 = rac{2x^2 + 2}{x^2 - 1} - 2 = 2\left( rac{x^2 + 1}{x^2 - 1} - 1

["Understanding the Simplified Form of a Key Algebraic Expression", "Mathematics often reveals elegant simplifications beneath complex-looking expressions. One such important rational function is:", "[\nf(x) = \frac{(x+1)^2 + (x-1)^2}{(x-1)(x+1)} - 2\n]", "This seemingly intricate expression simplifies beautifully to a cleaner rational form—critical for solving equations, analyzing domain restrictions, and applying calculus. In this article, we explore the step-by-step simplification of ( f(x) ), demonstrating how algebraic manipulation leads to a much more usable expression, and clarifying the domain considerations essential for real-world and theoretical applications.", "---", "### Step 1: Expand the Numerator", "Start with the original function:", "[\nf(x) = \frac{(x+1)^2 + (x-1)^2}{(x-1)(x+1)} - 2\n]", "First, expand the squared terms in the numerator:", "[\n(x+1)^2 = x^2 + 2x + 1\n]\n[\n(x-1)^2 = x^2 - 2x + 1\n]", "Add them together:", "[\n(x+1)^2 + (x-1)^2 = (x^2 + 2x + 1) + (x^2 - 2x + 1) = 2x^2 + 2\n]", "So the function becomes:", "[\nf(x) = \frac{2x^2 + 2}{(x-1)(x+1)} - 2\n]", "---", "### Step 2: Simplify the Denominator", "Note that:", "[\n(x-1)(x+1) = x^2 - 1\n]", "So now:", "[\nf(x) = \frac{2x^2 + 2}{x^2 - 1} - 2\n]", "Factor numerator:", "[\n2x^2 + 2 = 2(x^2 + 1)\n]", "Thus:", "[\nf(x) = \frac{2(x^2 + 1)}{x^2 - 1} - 2\n]", "---", "### Step 3: Combine into a Single Fraction", "Write ( -2 ) as a fraction with denominator ( x^2 - 1 ):", "[\nf(x) = \frac{2(x^2 + 1) - 2(x^2 - 1)}{x^2 - 1}\n]", "Simplify the numerator:", "[\n2(x^2 + 1) - 2(x^2 - 1) = 2x^2 + 2 - 2x^2 + 2 = 4\n]", "So:", "[\nf(x) = \frac{4}{x^2 - 1}\n]", "---", "### Step 4: Final Simplified Form", "P одиnote but return to the requested form. To match the target simplification:", "[\nf(x) = \frac{2x^2 + 2}{x^2 - 1} - 2 = 2\left( \frac{x^2 + 1}{x^2 - 1} - 1 \right)\n]", "This form is especially useful when analyzing limits, asymptotes, or performing calculus operations, because it separates the function into a rational component and a constant shift. Verify it:", "[\n2\left( \frac{x^2 + 1}{x^2 - 1} - 1 \right) = 2\left( \frac{x^2 + 1 - (x^2 - 1)}{x^2 - 1} \right) = 2\left( \frac{2}{x^2 - 1} \right) = \frac{4}{x^2 - 1}\n]", "Confirmed!", "---", "### Domain Considerations", "When simplifying rational expressions, domain restrictions must not be overlooked. The original function:", "[\nf(x) = \frac{(x+1)^2 + (x-1)^2}{(x-1)(x+1)} - 2\n]", "has denominator ( (x-1)(x+1) ), which is zero when ( x = \pm 1 ). Therefore, ( x <br/>\neq 1 ) and ( x <br/>\neq -1 ).", "In the simplified form:", "[\nf(x) = \frac{4}{x^2 - 1}\n]", "the same restrictions apply: ( x <br/>\neq \pm 1 ), because the denominator becomes zero here.", "---", "### Practical Implications and Applications", "This simplified expression is more tractable in various contexts:", "- Graphing: Visualizing vertical asymptotes at ( x = \pm 1 ) and horizontal asymptote at ( y = 0 ).\n- Calculus: Easier to compute limits, derivatives, and integrals.\n- Partial Fractions: The expression ( \frac{2x^2 + 2}{x^2 - 1} ) can be decomposed more effectively once simplified.\n- Equation Solving: Solving ( f(x) = k ) becomes straightforward.", "---", "### Conclusion", "The journey from the original complex fraction:", "[\nf(x) = \frac{(x+1)^2 + (x-1)^2}{(x-1)(x+1)} - 2\n]", "to the compact, insightful form\n[\n\boxed{f(x) = 2\left( \frac{x^2 + 1}{x^2 - 1} - 1 \right)}\n]", "demonstrates the power of algebraic simplification. Recognizing these transformations strengthens problem-solving skills and deepens mathematical understanding, whether in algebra, calculus, or applied fields.", "---", "Keywords:\n( f(x) = \frac{(x+1)^2 + (x-1)^2}{(x-1)(x+1)} - 2 ), simplify algebraic expression, rational function, domain restrictions, calculus readiness, algebra simplification, ( x^2 - 1 ), partial fractions, function analysis", "---", "Understanding these elegant transformations opens doors to more advanced mathematics—empowering students, educators, and professionals alike."]