Question

Найдите область значений функций:

Answer 1

У дроби знаменатель не равняется 0
sinx≠0   x≠πn n∈Z
cosx≠0  x≠π/2+πk k∈Z
f(x)=sin^2x/sinx + cos^2/cosx=sinx+cosx
берем производную и приравниваем 0 (ищем экстремумы)
f'(x)=cosx-sinx=0
cosx=sinx
tgx=1 (или ctgx=1 cosx≠0 sinx≠0)
x=π/4+2πn
x=5π/4+2πk
[π/4+2πn 5π/4+2πn] 
максимум
f(y)=sinx+cosx=√2/2+√2/2=√2
минимум
f(x)=-√2/2-√2/2=-√2
накладываем одз πn и π/2+πn попадает π/2 и π 
cos π/2 + sin π/2 = 0 +1=1
cos π + sin π = -1 +0 =-1
Область значений [-√2  √2]
накладываем одp [-√2 -1)U (-1 1) U (1 √2]

Answer 2

Finding the Range of Functions

To find the range of functions, we need to analyze the given functions and determine the set of possible output values. Let's examine each function individually:

1. Function 1: y = 1 - 3tg^2 x - The range of this function is not explicitly mentioned in the search results. - We cannot determine the exact range without additional information.

2. Function 2: y = cos^2x + cosx + 2 - The range of this function is not explicitly mentioned in the search results. - We cannot determine the exact range without additional information.

3. Function 3: y = -4cosx - The search results indicate that the range of this function is [-4, 4]. - Therefore, the possible output values of this function range from -4 to 4.

4. Function 4: y = 3x + 2 - The search results do not provide information about this specific function. - However, the function is a linear function with a slope of 3 and a y-intercept of 2. - Therefore, the range of this function is all real numbers.

5. Function 5: y = log nk - The search results do not provide information about this specific function. - However, the function represents a logarithmic function with a base of n and an input value of k. - The range of a logarithmic function depends on the base and the domain of the function. - Without additional information, we cannot determine the exact range.

Based on the available information, we can determine the range for some of the given functions. However, for functions without explicit range information, we cannot provide a definitive answer without additional context or calculations.