Question

The graph of the function f(x) = (x − 3)(x + 1) is shown.

On a coordinate plane, a parabola opens up. It goes through (negative 1, 0), has a vertex at (1, negative 4), and goes through (3, 0).
Which describes all of the values for which the graph is positive and decreasing?

all real values of x where x < −1
all real values of x where x < 1
all real values of x where 1 < x < 3
all real values of x where x > 3

Answer 1

We need values where f(x) > 0 and f is decreasing.

f(x) = (x−3)(x+1) is positive for x < −1 or x > 3 (outside the roots −1 and 3).
f'(x) = 2(x−1), so f is decreasing when x < 1.

Intersection: (x < −1) ∩ (x < 1) = x < −1.

Answer: all real values of x where x < −1.