To graph the solution set for the inequalities involving the functions f(x) = x^2 + 3 and g(x) = x^2 + 2, we should first interpret the given functions as inequalities. Typically, we want to determine where one function is greater than or less than the other, leading to a system of inequalities such as:
- f(x) > g(x)
- f(x) < g(x)
To modify the graphs accordingly, start by plotting the two functions on the same set of axes. You’ll observe that both functions are parabolas opening upwards. The vertex of f(x) is at (0, 3), while the vertex of g(x) is at (0, 2).
Next, to graph the solution set:
- Determine intersections: Find the points where f(x) and g(x) intersect by setting x^2 + 3 = x^2 + 2, which simplifies to 3 = 2. Since this is not true for any x, the graphs do not intersect.
- Identify regions: Since f(x) is always above g(x), the solution set for the inequality f(x) > g(x) includes the entire region above the line defined by g(x).
- Shade the appropriate regions: Shade the area above the graph of g(x) indicating all the points (x, y) such that y > g(x) and y > f(x) in the context of whichever inequalities you are working with.
To summarize, by identifying critical points and shading the relevant regions for each inequality, you effectively represent the solution set graphically. The solution set is represented graphically as the areas where the conditions of these inequalities hold true.