Then calculate $ f(1) = 3(1)^2 + 2(1) + 1 = 3 + 2 + 1 = 6 $.

["Understanding Linear and Quadratic Functions: Calculating $ f(1) $ with $ f(x) = 3x^2 + 2x + 1 $", "When studying functions in mathematics, evaluating a function at a specific input is a fundamental skill. One simple yet insightful example is calculating $ f(1) $ for the quadratic function:", "[\nf(x) = 3x^2 + 2x + 1\n]", "By substituting $ x = 1 $ into the expression, we reveal how the function behaves at this key point. Let’s walk through the process step-by-step and uncover the value.", "### Step-by-Step Calculation", "Start with the function:", "[\nf(x) = 3x^2 + 2x + 1\n]", "Now substitute $ x = 1 $:", "[\nf(1) = 3(1)^2 + 2(1) + 1\n]", "First, calculate the square:", "[\n(1)^2 = 1\n]", "Next, multiply:", "[\n3(1) = 3 \quad \ ext{and} \quad 2(1) = 2\n]", "Now add all terms:", "[\nf(1) = 3 + 2 + 1 = 6\n]", "### Why This Matters", "This result, $ f(1) = 6 $, shows the output of the function when the input is 1. In algebra and calculus, evaluating functions at specific points helps analyze behavior, find roots, and understand trends. For quadratic functions like this, evaluating $ f(1) $ gives insight into the function’s rise from the origin and its general upward U-shape due to the positive coefficient of $ x^2 $.", "### Summary", "- Function: $ f(x) = 3x^2 + 2x + 1 $\n- Evaluate at $ x = 1 $: $ f(1) = 3(1)^2 + 2(1) + 1 $\n- Calculation:\n $ 3(1) = 3 $,\n $ 2(1) = 2 $,\n $ 3 + 2 + 1 = 6 $\n- Final value: $ f(1) = 6 $", "Understanding how to compute function values is essential for mastering algebraic expressions and visualization of graphs. Try it yourself with different inputs—like $ f(0) $ or $ f(-2) $—to deepen your grasp of quadratic behavior.", "Keywords: evaluate $ f(1) $, quadratic function $ f(x) = 3x^2 + 2x + 1 $, function calculation, algebra tutorial, math learning, quadratic evaluation, solve for $ f(1) $, function example."]