Question:** A palynologist is analyzing pollen distribution in a region and models the concentration as a function \(f(x) = \frac{2x^2 - 3x + 1}{x^2 + 1}\). Determine the range of \(f(x)\) as \(x\) varies over all real numbers.

["Understanding the Range of Pollen Concentration Modeled by ( f(x) = \frac{2x^2 - 3x + 1}{x^2 + 1} )", "A palynologist studying pollen dispersion across a region often uses mathematical models to understand spatial patterns and concentrations. One such model is given by the rational function:", "[\nf(x) = \frac{2x^2 - 3x + 1}{x^2 + 1}\n]", "To support accurate ecological interpretation, determining the range of this function—i.e., all possible values of ( f(x) ) as ( x ) spans all real numbers—is essential. This article guides you through the analytical steps to find the range of ( f(x) ), explaining key concepts and techniques used.", "---", "### What Does the Range Represent in Palynology?", "In palynological analysis, ( f(x) ) might represent pollen concentration at different distances or locations. The range tells us the full spectrum of possible concentration values in the modeled region, helping scientists assess variability, identify thresholds, and inform conservation or sampling strategies.", "---", "### Step 1: Analyze the Function Structure", "We consider:", "[\nf(x) = \frac{2x^2 - 3x + 1}{x^2 + 1}\n]", "The denominator ( x^2 + 1 ) is always positive for all real ( x ), so the function is defined everywhere, with no vertical asymptotes or domain restrictions.", "Since both numerator and denominator are quadratic polynomials, and the degrees are equal, the horizontal asymptote determines the limiting behavior as ( x \ o \pm\infty ):", "[\n\lim_{x \ o \pm\infty} f(x) = \lim_{x \ o \pm\infty} \frac{2x^2}{x^2} = 2\n]", "So, ( f(x) \ o 2 ) as ( |x| ) increases — this suggests 2 is a horizontal asymptote, and the function may approach but not exceed this value far from the origin.", "---", "### Step 2: Solve for ( y = f(x) ) and Find Feasible Values", "To find the full range, set:", "[\ny = \frac{2x^2 - 3x + 1}{x^2 + 1}\n]", "Multiply both sides by ( x^2 + 1 ) (which is never zero):", "[\ny(x^2 + 1) = 2x^2 - 3x + 1\n]", "Expand the left side:", "[\nyx^2 + y = 2x^2 - 3x + 1\n]", "Bring all terms to one side:", "[\nyx^2 + y - 2x^2 + 3x - 1 = 0\n]", "Group like terms:", "[\n(y - 2)x^2 + 3x + (y - 1) = 0\n]", "This is a quadratic equation in ( x ). For real ( x ) to exist, the discriminant must be non-negative:", "[\n\Delta = b^2 - 4ac \geq 0\n]", "Here, ( a = y - 2 ), ( b = 3 ), ( c = y - 1 ). So:", "[\n\Delta = 3^2 - 4(y - 2)(y - 1) \geq 0\n]", "[\n9 - 4(y - 2)(y - 1) \geq 0\n]", "Now expand ( (y - 2)(y - 1) ):", "[\n(y - 2)(y - 1) = y^2 - 3y + 2\n]", "So:", "[\n9 - 4(y^2 - 3y + 2) \geq 0\n]", "[\n9 - 4y^2 + 12y - 8 \geq 0\n]", "[\n-4y^2 + 12y + 1 \geq 0\n]", "Multiply both sides by (-1) (reversing the inequality):", "[\n4y^2 - 12y - 1 \leq 0\n]", "---", "### Step 3: Solve the Quadratic Inequality", "We solve ( 4y^2 - 12y - 1 = 0 ) using the quadratic formula:", "[\ny = \frac{12 \pm \sqrt{(-12)^2 - 4(4)(-1)}}{2(4)} = \frac{12 \pm \sqrt{144 + 16}}{8} = \frac{12 \pm \sqrt{160}}{8}\n]", "Simplify ( \sqrt{160} = \sqrt{16 \cdot 10} = 4\sqrt{10} ), so:", "[\ny = \frac{12 \pm 4\sqrt{10}}{8} = \frac{3 \pm \sqrt{10}}{2}\n]", "Let:", "[\ny_1 = \frac{3 - \sqrt{10}}{2}, \quad y_2 = \frac{3 + \sqrt{10}}{2}\n]", "Since the parabola ( 4y^2 - 12y - 1 ) opens upwards, the expression ( 4y^2 - 12y - 1 \leq 0 ) holds between the roots:", "[\n\frac{3 - \sqrt{10}}{2} \leq y \leq \frac{3 + \sqrt{10}}{2}\n]", "---", "### Step 4: Verify Achievability of Endpoints", "We confirm that both bounds are attainable because the discriminant is zero at each endpoint, meaning there is exactly one real ( x ) satisfying the equation — so each extreme value is achieved.", "At ( x = 0 ), ( f(0) = \frac{1}{1} = 1 ), which lies within ( \left[\frac{3 - \sqrt{10}}{2}, \frac{3 + \sqrt{10}}{2}\right] ), confirming part of the interval is valid. The endpoints correspond to locally maximum/minimum behavior under the asymptotic constraint.", "---", "### Final Interpretation: The Range of Pollen Concentration", "The function ( f(x) = \frac{2x^2 - 3x + 1}{x^2 + 1} ) achieves all real values between ( \frac{3 - \sqrt{10}}{2} ) and ( \frac{3 + \sqrt{10}}{2} ), inclusive.", "Thus, the range of this palynological model is:", "[\n\left[ \frac{3 - \sqrt{10}}{2}, \frac{3 + \sqrt{10}}{2} \right]\n]", "This interval defines the complete spectrum of plausible pollen concentrations across the modeled region, providing a solid mathematical foundation for ecological inference and sampling design.", "---", "### Conclusion", "By analyzing the rational function modeling pollen distribution, we used algebraic methods—specifically discriminant analysis of a quadratic equation—to determine the full range of concentrations. This approach bridges palynology and mathematical modeling, enabling precise interpretation of environmental data.", "For researchers and educators, understanding such functional ranges enhances the accuracy of spatial modeling in palynology, supporting informed decisions in environmental monitoring and biodiversity assessment.", "---", "Keywords: palynology, pollen distribution, rational function, function range, ( f(x) = \frac{2x^2 - 3x + 1}{x^2 + 1} ), asymptote, discriminant method, real solutions, mathematical modeling, ecological concentration."]