The equation
f(x)=k gives the solutions to the system of equations
y=f(x)=x3−x2−x− and
y=k A real solution of a system of two equations corresponds to a point of intersection of the graphs of the two equations in the xy-plane. The graph of
y=k is a horizontal line thatcontains the point
(0,k). Thus, the line with equation
y=−3 is a horizontal line that intersects the graph of the cubic equation three times, and it follows that the equation
f(x)=−3 has three real solutions.
Choices A, B, and C are incorrect because the graphs of the corresponding equations are horizontal lines that do not intersect the graph of the cubicequation three times.