Solution: To determine the value of $ k $, substitute $ x = 3 $ into the function and set $ f(3) = 0 $:

["Solution Guide: How to Determine the Value of ( k ) by Substituting and Setting ( f(3) = 0 )", "When solving equations involving functions, particularly linear or function models, a powerful technique is to substitute known values into the function expression and use given conditions to isolate unknown parameters. One common method is to substitute ( x = 3 ) into the function and set ( f(3) = 0 ). This approach helps determine useful constants like ( k ), which often define critical points such as roots, intercepts, or thresholds in real-world applications. In this article, we’ll explore how to properly apply this solution technique step-by-step and why it’s effective.", "---", "### Understanding the Concept", "In many algebraic and applied math problems, functions are defined with an unknown constant—often labeled ( k )—that determines behavior such as crossing the x-axis or reaching a specific output. To pinpoint ( k ), we use known input-output pairs. Substituting ( x = 3 ) into ( f(x) ) is a strategic choice when we know ( f(3) ), enabling us to form an equation ( f(3) = 0 ) if the function is expected to satisfy ( f(3) = 0 ). This sets the stage for linear algebra or equation solving.", "---", "### When Is This Method Used?", "This substitution technique is especially valuable in:", "- Linear function analysis: When working with functions like ( f(x) = mx + b ), or more complex forms involving a constant term ( k ), establishing ( f(c) = d ) helps find the missing constant.", "- Equations requiring root identification: If solving for ( x ) when ( f(x) = 0 ), substituting specific ( x ) values allows evaluating known outputs and isolating ( k ).", "- Model fitting in applied math: In physics, economics, or engineering, knowing a model’s behavior at specific inputs aids calibration using real data.", "---", "### Step-by-Step Solution: Solving for ( k ) with ( f(3) = 0 )", "Let’s apply the method with a generic linear function containing ( k ), then show how substitution leads to solving for ( k ).", "Step 1: Define the function\nSuppose ( f(x) = 2x + k ). Here, ( k ) is the unknown constant.", "Step 2: Use the condition ( f(3) = 0 )\nSubstitute ( x = 3 ) into the function:\n[\nf(3) = 2(3) + k = 6 + k\n]", "According to the condition:\n[\nf(3) = 0 \Rightarrow 6 + k = 0\n]", "Step 3: Solve for ( k )\n[\nk = -6\n]", "Thus, the value ( k = -6 ) ensures the function passes through the point ( (3, 0) ).", "---", "### Practical Example", "Imagine modeling projected revenue:\n[\nf(x) = -10x + k\n]\nIf at ( x = 3 ) months, the revenue is predicted to be zero, set:\n[\nf(3) = -10(3) + k = -30 + k = 0\n\Rightarrow k = 30\n]\nNow we know ( k = 30 ), so the full function is ( f(x) = -10x + 30 ), and note that ( f(3) = 0 )—revenue breaks even at 3 months.", "---", "### Why This Method Works", "Substituting ( x = 3 ) provides a concrete equation involving ( k ), turning an unknown into an algebraic solvable variable. By aligning function values with real conditions (e.g., zero output), the math becomes concrete and actionable—critical for academic problems and professional modeling.", "---", "### Summary", "Determining ( k ) by substituting ( x = 3 ) and setting ( f(3) = 0 ) is a clear, efficient method rooted in functional evaluation and algebraic manipulation. Whether for theory or application, this approach enables precise identification of constants that dictate function behavior. Use it when you have known input-output pairs and target values to unlock unknown parameters.", "---", "Key Takeaways:", "- Substitute the given ( x )-value into ( f(x) ).\n- Apply the known condition ( f(3) = 0 ) to form an equation.\n- Solve algebraically for the constant ( k ).\n- This method bridges function behavior with real-world or theoretical requirements.", "By mastering this technique, students and professionals alike gain a valuable tool for solving k-based equations confidently and accurately."]