f(-1.5) = -1.5 - \frac{(-1.5)^2}{2} = -1.5 - \frac{2.25}{2} = -1.5 - 1.125 = -2.625
["Understanding f(x) = x - \frac{x²}{2} at x = -1.5: A Step-by-Step Breakdown", "When exploring mathematical functions, evaluating expressions at specific values offers clear insights into their behavior. One such function of interest is:", "[\nf(x) = x - \frac{x^2}{2}\n]", "Understanding how this function evaluates at ( x = -1.5 ) not only deepens our grasp of algebra but also highlights useful computational techniques. Let’s walk through the calculation of ( f(-1.5) ) step by step.", "---", "### The Function: A Simple Quadratic Expression", "The function ( f(x) = x - \frac{x^2}{2} ) combines a linear term ( x ) and a quadratic penalty ( -\frac{x^2}{2} ). This hybrid form appears in various contexts, such as approximations in calculus and physics models of motion with resistance or damping.", "We are asked to compute:", "[\nf(-1.5) = -1.5 - \frac{(-1.5)^2}{2}\n]", "---", "### Step-by-Step Evaluation", "Step 1: Square the input value", "First, compute ( (-1.5)^2 ):", "[\n(-1.5)^2 = (-1.5) \ imes (-1.5) = 2.25\n]", "Even though the input is negative, squaring eliminates the sign—important to remember when working with even powers.", "Step 2: Divide the square by 2", "Next, divide the result by 2:", "[\n\frac{2.25}{2} = 1.125\n]", "Step 3: Subtract this value from ( x )", "Now plug into the function:", "[\nf(-1.5) = -1.5 - 1.125 = -2.625\n]", "---", "### Final Result", "Thus,", "[\nf(-1.5) = -2.625\n]", "This value represents the output of the function at ( x = -1.5 ), combining both linear and quadratic dependencies on the input.", "---", "### Why This Calculation Matters", "Understanding such evaluations helps build fluency in algebraic manipulation and function analysis. It also lays groundwork for:", "- Numerical approximations: The formula resembles Taylor expansions used in calculus, where approximations of functions involve polynomial terms.\n- Graphical interpretation: Knowing ( f(-1.5) ) allows plotting this point to visualize the function’s shape.\n- Error estimation: In applied mathematics, expressions like ( \frac{x^2}{2} ) often simulate real-world factors (e.g., resistance or friction), making evaluation crucial in modeling.", "---", "### Summary", "Evaluating ( f(x) = x - \frac{x^2}{2} ) at ( x = -1.5 ) yields:", "[\nf(-1.5) = -2.625\n]", "Through careful step-by-step calculation—squaring, dividing, and subtracting—we gain clarity on function behavior and strengthen core mathematical reasoning. Whether for homework, exams, or applied problems, mastering these procedures empowers deeper insight and confidence in working with mathematical functions.", "---", "Key takeaways:\n- Always compute powers before applying coefficients.\n- Even with negative inputs, squaring removes negative signs.\n- Breaking down expressions step by step reduces errors and improves understanding.", "---", "#### Needs further examples or advanced function analysis?\nExplore how polynomial functions behave across domains or learn how quadratic terms model real-world phenomena—continuing your journey in algebra and applied math starts here!"]
