The figure above shows the graph in the xy-plane of the function f. If q, r, s and t are distinct real numbers, which of the following could be f(?
A. f(x) = (x - q)2
B. f(x) = (x - r)(x + s)
C. f(x) = (x - r)(x + s)(x + t)
D. f(x) = (x - q)(x - r)(x + s)(x + t)
Answer:C
The figure above shows the graph in the xy-plane of the function f. If q, r, s and t are distinct real numbers, which of the following could be f(?
A. f(x) = (x - q)2 B. f(x) = (x - r)(x + s) C. f(x) = (x - r)(x + s)(x + t) D. f(x) = (x - q)(x - r)(x + s)(x + t)
Explanation: C The graph crosses the x-axis at three distinct points. When the function is set to 0, there should be three real solutions for x. Use Process of Elimination to solve this question. Set the equation in (A) to 0 to get 0 = (x - q)2. In this equation, the root is at x = q, thereby providing only one real value for x. Eliminate (A). Set the equation in (B) to 0 to get 0 = (x - q)(x + s). The solutions for this equation are x = q or x = -s. Therefore, there are only two real solutions for x. Eliminate (B). Set the equation in (C) to 0 to get 0 = (x - r)(x + s)(x + t). The solutions for this equation are x = r, x = -s, and x = -t. Therefore, there are three real solutions for x. The correct answer is (C).