## Chart/Diagram Type: Math Problem with Graph ### Overview The image presents a math problem involving two functions, f(x) and g(x). It asks how many roots the functions have in common. The image includes the equation for f(x), a graph of g(x), and a step-by-step solution to the problem. ### Components/Axes * **Title:** Data Example from MathVerse * **Problem Statement:** * f(x) = x² + x - 6. How many roots do the functions have in common, f(x) and g(x)? * Choices: A: f and g share the same root(s). B: f and g share one root in common but each have another root that is not shared. C: f and g share no roots in common. * **Graph:** * Axes: x-axis and y-axis * x-axis: Ranges from -7 to 7, with integer markings. * y-axis: Ranges from -7 to 7, with integer markings. * Curve: A parabola representing g(x), labeled as "y = g(x)". The parabola intersects the x-axis at x = -3 and x = 4. The vertex of the parabola appears to be around (0.5, -2.25). * **Solution:** * Step-by-step reasoning to find the roots of f(x) and g(x) and compare them. * Final answer indicating that f(x) and g(x) share one root in common but each have another root that is not shared. ### Detailed Analysis or ### Content Details * **Function f(x):** f(x) = x² + x - 6 * **Roots of f(x):** The solution shows that f(x) can be factored into (x + 3)(x - 2), so the roots are x = -3 and x = 2. * **Graph of g(x):** The parabola intersects the x-axis at x = -3 and x = 4. Therefore, the roots of g(x) are x = -3 and x = 4. * **Comparison:** Both functions share the root x = -3. f(x) has a root at x = 2, while g(x) has a root at x = 4. * **Solution Steps:** * Step-0 (Neural) * Step-1 (Correct) * Step-2 (Incorrect) * Step-3 (Incorrect) * Step-4 (Incorrect) * **Answer:** The correct option is B. f and g share one root in common but each have another root that is not shared. ### Key Observations * The graph of g(x) is a parabola opening upwards. * The roots of f(x) are found algebraically, while the roots of g(x) are determined from the graph. * The solution correctly identifies the shared root and the unique roots for each function. ### Interpretation The problem demonstrates how to find the roots of a quadratic function algebraically and graphically. By comparing the roots of f(x) and g(x), the solution determines the number of common roots and identifies the correct answer choice. The step-by-step reasoning provides a clear explanation of the solution process. The "incorrect" labels on steps 2-4 suggest that these steps might represent common errors or alternative approaches that do not lead to the correct answer.