Nonlinear Functions

Question

Which of the following is an example of a function whose graph in the xy-plane has exactly two x-intercepts?



A. A linear function whose rate of change is zero

B. A quadratic function with no solutions

C. A quadratic function with multiple solutions

D. A cubic polynomial with one solution

Answer 1

C is correct or the best answer, if it means multiple real solutions. Every x intercept is a real solution, unless the x intercept is imaginary. If the graph of the quadratic crosses the x axis twice, it has two real solutions. Even if it just touches the x axis with just one real solution that solution has multiplicity 2, and is really two solutions, just repetitive, as in y= (x-2)^2 = (x-2)(x-2) has two solutions: 2 and 2, both the same.

Go with C

B is definitely wrong. It's a parabola completely below or above the x axis, never intersecting the x axis

A is wrong. It is either a line coexistent with the x axis, with infinite solutions or it's a line parallel to the x axis with no intersections, no solutions.

D is wrong. It's a cubic, which must intersect the x axis at least once, maybe twice or thrice. But not twice except when one solution has multiplicity 2, which really is 3 solutions Cubics have 3 solutions, with either 3 real solutions or 1 solution and 2 imaginary, or 2 solutions both real, but one repetitive, multiplicity 2. in any event cubics do not have exactly 2 solutions except rarely.