Question

18)
a) increasing: (-0.7, 0), (0.7, ∞) decreasing: (-∞, -0.7), (0, 0.7)
b) increasing: (-∞, -1) decreasing: (-1, ∞)
c) increasing: (-∞, -2), (-2, ∞) decreasing: no intervals exist.
d) increasing: (-∞, -3) decreasing: (-3, ∞)

Explanation:

Step1: Analyze the function's trend

For a function \(y = f(x)\), if the function is increasing, as \(x\) increases, \(y\) also increases. If the function is decreasing, as \(x\) increases, \(y\) decreases.

Step2: Locate the vertex

The vertex of the parabola (since it's a quadratic - like graph) is the point where the function changes from increasing to decreasing. From the graph, the vertex (the highest point of the parabola) occurs at \(x=-1\).

Step3: Determine increasing and decreasing intervals

For \(x < - 1\), as \(x\) increases (moving from left to right on the \(x\) - axis), the \(y\) - values of the function increase. So the function is increasing on the interval \((-\infty,-1)\).
For \(x>-1\), as \(x\) increases (moving from left to right on the \(x\) - axis), the \(y\) - values of the function decrease. So the function is decreasing on the interval \((-1,\infty)\).

Option A is incorrect because the intervals are mis - identified. Option C is incorrect as a parabola (a non - linear function in this case) cannot be increasing on two disjoint intervals \((-\infty,-2)\) and \((-2,\infty)\) simultaneously. Option D is incorrect because the vertex is not at \(x =-3\) (from the graph).

Answer:

B. Increasing: \((-\infty, - 1)\) Decreasing: \((-1,\infty)\)