A function f(x) is defined as follows: f(x) = 2x if x < 0, f(x) = x^2 if 0 <= x <= 1, and f(x) = x if x > 1. How many times does the graph of y = f(x) intersect the line y = x?
Correct: C
To find the number of intersections between the graph of y = f(x) and the line y = x, we need to consider the three different cases for f(x) and find the points of intersection for each case. For x < 0, we have f(x) = 2x, so we are looking for points where 2x = x. This equation has only one solution, x = 0, but since x < 0, there are no solutions in this case. For 0 <= x <= 1, we have f(x) = x^2, so we are looking for points where x^2 = x. This equation has two solutions, x = 0 and x = 1. Both of these points are in the interval 0 <= x <= 1, so both are valid solutions. For x > 1, we have f(x) = x, so we are looking for points where x = x. This equation has an infinite number of solutions, but they are all in the interval x > 1. However, the line y = x intersects the graph of y = f(x) only once in this interval, at the point (1, 1), which is the boundary between the interval 0 <= x <= 1 and the interval x > 1. However, looking at our function definition for the case when x > 1 we have a linear equation. Thus our graph of y = f(x) will have 2 intersections with y = x.