A math competition problem asks: If \( f(x) = 3x^2 - 4x + 7 \) and \( g(x) = 2x - 5 \), what is the value of \( f(g(3)) \)?

["# How to Solve Composition of Functions: A Classic Math Competition Problem", "If you've ever encountered a problem like: Given ( f(x) = 3x^2 - 4x + 7 ) and ( g(x) = 2x - 5 ), what is ( f(g(3)) )? — you're not alone. This type of composition of functions is a staple in algebra and often appears in math competitions due to its clear structure and demand for logical precision and step-by-step reasoning.", "In this article, we’ll explore how to compute ( f(g(3)) ) efficiently, breaking down the process with thorough explanation ideal for students preparing for math olympiads or competitive exams. Mastering this concept strengthens foundational skills used across higher-level math topics.", "### What Does Composing Functions Mean?", "Function composition ( f(g(x)) ) means applying one function to the result of another. In this case, we first evaluate ( g(3) ), then plug the result into ( f(x) ).", "### Step-by-Step Solution", "#### Step 1: Compute ( g(3) )", "Start with the inner function ( g(x) = 2x - 5 ).\nSubstitute ( x = 3 ):", "[\ng(3) = 2(3) - 5 = 6 - 5 = 1\n]", "So, ( g(3) = 1 ).", "#### Step 2: Plug ( g(3) = 1 ) into ( f(x) )", "Now evaluate ( f(g(3)) = f(1) ) using ( f(x) = 3x^2 - 4x + 7 ):", "[\nf(1) = 3(1)^2 - 4(1) + 7 = 3(1) - 4 + 7 = 3 - 4 + 7 = 6\n]", "#### Final Answer", "[\nf(g(3)) = 6\n]", "---", "### Why This Problem Appears in Competitions", "1. Syllabus Integration: Composition of functions is a core algebra concept tested to assess problem-solving fluency and algebraic manipulation.\n2. Precision & Clear Steps: Competitions value logical clarity — showing each step ensures correctness and demonstrates deep understanding.\n3. Foundation for More Complex Problems: This simple task builds intuition for nested operations, transformations, and applied modeling in advanced math.", "### Tips for Tackling Similar Problems", "- Always start with the inner function: Compute values step-by-step before substituting back.\n- Substitute carefully: Double-check your input values match the function definitions exactly.\n- Label intermediate steps: This minimizes errors, especially in timed competitions.\n- Verify units and expressiveness: In applied problems, interpret results meaningfully beyond just getting 6.", "---", "If you're sharpening your algebra skills for contests, problems like this aren’t just exercises — they’re practice for conquering complex, layered challenges. Mastering function composition opens the door to powerful techniques in calculus, optimization, and numerical modeling.", "Keep practicing — and remember, every function nest holds the key to the next insight!", "---", "Keywords: function composition, math competition problem, algebra, solving f(g(x)), how to compute f(g(x)), step-by-step math, quadratic functions, math strategy, problem-solving skills"]