F(x) = 3x+2, x≤2 f(x) = -0,5x-3, -2

Question 1

Задача Задача №68445

А

Аноним

28 ноября 2023 21:35 · публичный вопрос

А

Аноним 28 ноября 2023 21:35

F(x) = 3x+2, x≤2 f(x) = -0,5x-3, -2

Answer

D(f) - definition of the domain of the function f E(f) - range (or set of possible output values) of the function f Zeroes of the function - values of x for which f(x) = 0 Point of intersection with Oy - value of f(0) Increasing or decreasing function - check the slope or the sign of the derivative of the function

To determine the domain of the function f, we need to consider the given conditions for each part of the function separately:

For the first part (x ≤ 2), there are no restrictions on the domain.
For the second part (-2 < x < 0), there are no restrictions on the domain.
For the third part (x ≥ 0), there are no restrictions on the domain.

Therefore, the domain of the function f is (-∞, ∞).

To determine the range of the function f, we need to consider the possible output values for each part of the function separately:

For the first part (x ≤ 2), the output values of f(x) are all real numbers.
For the second part (-2 < x < 0), the output values of f(x) are all real numbers.
For the third part (x ≥ 0), the output values of f(x) are all real numbers.

Therefore, the range of the function f is (-∞, ∞).

To find the zeros of the function, we need to solve f(x) = 0 for each part of the function separately:

For the first part (x ≤ 2), solving 3x + 2 = 0 gives x = -2/3, which is within the domain of this part of the function.
For the second part (-2 < x < 0), solving -0.5x - 3 = 0 gives x = -6, which is within the domain of this part of the function.
For the third part (x ≥ 0), the function f(x) = -5 has no zeros.

Therefore, the zeros of the function are x = -2/3 and x = -6.

To find the point of intersection with the y-axis, we need to evaluate f(0) for each part of the function separately:

For the first part (x ≤ 2), f(0) = 3(0) + 2 = 2.
For the second part (-2 < x < 0), f(0) = -0.5(0) - 3 = -3.
For the third part (x ≥ 0), f(0) = -5.

Therefore, the point of intersection with the y-axis is (0, 2) for the first part of the function and (-3,0) for the second part of the function.

To determine if the function is increasing or decreasing, we need to examine the slope or the sign of the derivative of the function for each part separately:

For the first part (x ≤ 2), the function f(x) = 3x + 2 has a constant positive slope, indicating that it is increasing.
For the second part (-2 < x < 0), the function f(x) = -0.5x - 3 has a constant negative slope, indicating that it is decreasing.
For the third part (x ≥ 0), the function f(x) = -5 has a constant slope of 0, indicating that it is neither increasing nor decreasing.

Therefore, the function f is increasing for the first part, decreasing for the second part, and neither increasing nor decreasing for the third part.

Question 2

f(x) = 3x+2, x≤2 f(x) = -0,5x-3, -2

Answer

D(f) - definition of the domain of the function f E(f) - range (or set of possible output values) of the function f Zeroes of the function - values of x for which f(x) = 0 Point of intersection with Oy - value of f(0) Increasing or decreasing function - check the slope or the sign of the derivative of the function To determine the domain of the function f, we need to consider the given conditions for each part of the function separately: - For the first part (x ≤ 2), there are no restrictions on the domain. - For the second part (-2 < x < 0), there are no restrictions on the domain. - For the third part (x ≥ 0), there are no restrictions on the domain. Therefore, the domain of the function f is (-∞, ∞). To determine the range of the function f, we need to consider the possible output values for each part of the function separately: - For the first part (x ≤ 2), the output values of f(x) are all real numbers. - For the second part (-2 < x < 0), the output values of f(x) are all real numbers. - For the third part (x ≥ 0), the output values of f(x) are all real numbers. Therefore, the range of the function f is (-∞, ∞). To find the zeros of the function, we need to solve f(x) = 0 for each part of the function separately: - For the first part (x ≤ 2), solving 3x + 2 = 0 gives x = -2/3, which is within the domain of this part of the function. - For the second part (-2 < x < 0), solving -0.5x - 3 = 0 gives x = -6, which is within the domain of this part of the function. - For the third part (x ≥ 0), the function f(x) = -5 has no zeros. Therefore, the zeros of the function are x = -2/3 and x = -6. To find the point of intersection with the y-axis, we need to evaluate f(0) for each part of the function separately: - For the first part (x ≤ 2), f(0) = 3(0) + 2 = 2. - For the second part (-2 < x < 0), f(0) = -0.5(0) - 3 = -3. - For the third part (x ≥ 0), f(0) = -5. Therefore, the point of intersection with the y-axis is (0, 2) for the first part of the function and (-3,0) for the second part of the function. To determine if the function is increasing or decreasing, we need to examine the slope or the sign of the derivative of the function for each part separately: - For the first part (x ≤ 2), the function f(x) = 3x + 2 has a constant positive slope, indicating that it is increasing. - For the second part (-2 < x < 0), the function f(x) = -0.5x - 3 has a constant negative slope, indicating that it is decreasing. - For the third part (x ≥ 0), the function f(x) = -5 has a constant slope of 0, indicating that it is neither increasing nor decreasing. Therefore, the function f is increasing for the first part, decreasing for the second part, and neither increasing nor decreasing for the third part.