When 0

Draw a sketch of the function f(x) as shown in the figure.

The straight line y = kx+ 1 passes through the fixed point (0, 1). When it is tangent to y = -2/x, it is obtained by -2/x = kx+ 1

Kx 2+x+2 = 0, so the discriminant = 1-8k = 0 is k = 1/8.

Similarly, when a straight line is tangent to y = 2/x, k =-1/8.

It can be seen that when k > 1/8 or k

When-1/8

When k = 0 or-1/8 or 1/8, the straight line and the image of f(x) have exactly four intersections.

Answer: k = 0 or-1/8 or 1/8.