SAT Mathematics Practice Test 4

The equation $\mathit{f}\left(\mathit{x}\right)=\mathit{k}$ gives the solutions to the system of equations $\mathit{y}=\mathit{f}\left(\mathit{x}\right)={\mathit{x}}^3-{\mathit{x}}^2-\mathit{x}-\frac{11}{4}$ and $\mathit{y}=\mathit{k}$ 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 $\mathit{y}=\mathit{k}$ is a horizontal line thatcontains the point $\left(0,\mathit{k}\right)$. Thus, the line with equation $\mathit{y}=-3$ is a horizontal line that intersects the graph of the cubic equation three times, and it follows that the equation $\mathit{f}\left(\mathit{x}\right)=-3$ 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.