F(x) = 3x+2, x≤2 f(x) = -0,5x-3, -2

The given function, f(x), has different definitions for different intervals of x.

1. For x ≤ 2, f(x) = 3x + 2.
2. For -2 < x < 0, f(x) = -0.5x - 3.
3. For x ≥ 0, f(x) = -5.

Let's find the domain (D(f)), the range (E(f)), the zeros of the function, the point of intersection with the y-axis, and whether the function is increasing or decreasing.

1. Domain (D(f)): The domain of a function is the set of all possible values of x for which the function is defined. In this case, f(x) is defined for all real numbers.

D(f) = (-∞, ∞)

2. Range (E(f)): The range of a function is the set of all possible values of f(x) for the given domain.

For x ≤ 2, f(x) is a linear equation with a positive slope (3). As x increases, f(x) increases.

For -2 < x < 0, f(x) is a linear equation with a negative slope (-0.5). As x increases, f(x) decreases.

For x ≥ 0, f(x) is a constant function equal to -5. It does not depend on the value of x.

The range for each individual part can be determined separately.

For x ≤ 2, the range is (-∞, ∞) because it is a linear equation with positive slope and has no upper or lower bound.

For -2 < x < 0, the range is (-∞, -3) since the function decreases as x increases.

For x ≥ 0, the range is {-5} because the function is constant.

So, E(f) = (-∞, ∞) for x ≤ 2, (-∞, -3) for -2 < x < 0, and {-5} for x ≥ 0.

3. Zeros of the function: To find the zeros of the function, we need to find the values of x for which f(x) = 0.

For f(x) = 3x + 2, we set it equal to 0: 3x + 2 = 0 3x = -2 x = -2/3

There is no zero for the second part, f(x) = -0.5x - 3.

For f(x) = -5, there is no x that makes -5 = 0.

The zeros of the function occur at x = -2/3.

4. Intersection with y-axis: The point of intersection with the y-axis is the value of the function when x = 0.

For x ≤ 2, f(x) = 3x + 2, substituting x = 0: f(0) = 2

For -2 < x < 0, f(x) = -0.5x - 3, substituting x = 0: f(0) = -3

For x ≥ 0, f(x) = -5, substituting x = 0: f(0) = -5

The point of intersection with the y-axis is (0, 2) for x ≤ 2, (0, -3) for -2 < x < 0, and (0, -5) for x ≥ 0.

5. Increasing or decreasing function: For x ≤ 2, f(x) = 3x + 2. As x increases, f(x) also increases. So, the function is increasing in this interval.

For -2 < x < 0, f(x) = -0.5x - 3. As x increases, f(x) decreases. So, the function is decreasing in this interval.

For x ≥ 0, f(x) = -5. The function is constant, so it neither increases nor decreases.

In summary: D(f) = (-∞, ∞) E(f) = (-∞, ∞) for x ≤ 2, (-∞, -3) for -2 < x < 0, and {-5} for x ≥ 0 Zeros of the function: x = -2/3 Intersection with y-axis: (0, 2) for x ≤ 2, (0, -3) for -2 < x < 0, and (0, -5) for x ≥ 0 The function is increasing for x ≤ 2, decreasing for -2 < x < 0, and constant for x ≥ 0.