Question

Which statements correctly describe the graph of the function f(x) = x3 – 2x2 – 19x + 20? Select three options.

As the x-values increase, the y-values always increase.
As x approaches negative infinity, y approaches negative infinity.
The domain of the function is all real numbers.
The range of the function is y 20.
The graph has a positive y-intercept.

Answer 1

Correct answers: 2, 3, and 5.

Brief reasons:
- As x → −∞, x^3 → −∞, so y → −∞ (true).
- A polynomial has domain all real numbers (true).
- f(0) = 20, so the y-intercept is positive (true).
- The function is not always increasing (f'(x)=3x^2−4x−19 has real roots), so statement 1 is false.
- The range is not restricted to y ≥ 20 (a cubic has range all real numbers), so statement 4 is false.