Final answer:
To sketch the region bounded by the graphs of the functions f(x) = 3x - 3 and g(x) = x - 3, first plot both lines, find their intersection, and then shade the triangular region above g(x) and below f(x), to the right of the y-axis.
Step-by-step explanation:
The question asks us to sketch the region bounded by the graphs of the functions f(x) = 3x − 3 and g(x) = x − 3. To do this, we need to understand that both equations represent straight lines, where f(x) has a slope of 3 and g(x) has a slope of 1, and they both intersect the y-axis at −3. The region bounded by these two lines will form a triangular shape on a graph, where the vertical boundaries are defined by the points where the lines intersect and the horizontal lines are at the x-values that satisfy the condition given for each function.
To find the intersection point of the two lines, set f(x) = g(x):
- 3x − 3 = x − 3
- 2x = 0
- x = 0
Both lines intersect at the point (0, -3). Since f(x) is steeper, the region bounded by these two lines will exist to the right of the y-axis, and will be above g(x) and below f(x). This triangular region can be shaded on the graph, representing the area where the conditions f(x) > g(x) and x ≥ 0 are met.