Question

Найти область значений функции у=π/4-3arccos 2x:

Answer 1

Область значений функции arccos

D[arccos(2x)] = [0; π], то есть

0 ≤ arccos(2x) ≤ π, ⇔ 0 ≤ 3·arccos(2x) ≤ 3π, ⇔ 0 ≥ -3arccos(2x) ≥ -3π, ⇔

⇔ (π/4) ≥ (π/4) - 3arccos(2x) ≥ (π/4) - 3π = (π - 12π)/4 = -11π/4, ⇔

⇔ -11π/4 ≤ (π/4) - 3arccos(2x) ≤ π/4.

Ответ. [-11π/4; π/4].

Answer 2

Finding the Range of the Function y = π/4 - 3arccos(2x)

To find the range of the function y = π/4 - 3arccos(2x), we need to consider the range of the arccosine function and the transformation applied to it.

The range of the arccosine function is typically [0, π]. When a constant is multiplied to the arccosine function, it affects the amplitude of the resulting function. In this case, the amplitude is 3, and the function is shifted downward by π/4.

Range Calculation

The range of the function y = π/4 - 3arccos(2x) can be calculated as follows:

1. The amplitude of the arccosine function is 3, which means the range is stretched vertically by a factor of 3. 2. The function is shifted downward by π/4.

Resulting Range

The resulting range of the function y = π/4 - 3arccos(2x) is [-3π/4, 3π/4].

This range is obtained by considering the transformation applied to the arccosine function and the resulting shift in the range.

[[8 #]]