Prove the inequality  

Solve the inequality ||x| −2| ≤ 1

a) x > 3

b) − 3 ≤ x ≤ 3

c) 1 ≤ x ≤ 3

d) – 3 ≤ x ≤ − 1 or 1 ≤ x ≤ 3

Solve the inequality ||2 – 3x| −1| > 2

a)

b)

c)

d)  or

Find the domain of definition of the function
 

a) {0}

b) {0} ∪ [1, ∞)

c) (0, ∞)

d) [0, ∞)

Find the domain of definition of the function
 

a) x ≥ 0

b) 0 ≤ x ≤ 2nπ

c) 0 ≤ x ≤ (2nπ)²

d) (2nπ)² ≤ x ≤((2n + 1)π)², n = 0, 1, 2.  .  . 

Find the domain of definition of the function
 

a) x = 0

b) x = 0, 1, 2.  .  .

c) x = 0, ±1, ±2, .  .  .

d) Defined for all x except x = 0, ±1, ±2, .  .  .

Find the domain of definition of the function
 

a) {0}

b) (0, ∞)

c) (−∞, 0)

d) (−∞, ∞)

Find the domain of definition of the function
 

a) {0}

b) (0, ∞)

c) (−∞, 0)

d) Not defined for any x

Find the domain of definition of the function
 

a) [−4, −2]

b) [−4, 2]

c) [−4, 2] ∪ {4}

d) [−4, −2] ∪ [2, 4]

Find the domain of definition of the function
 

a) x = 0, π

b) x = nπ

c)

d) x = 2nπ; n = 0, ±1, ±2,.  . .  .

Are the following functions identical
 

a) Identical

b) Not identical

c) Cannot say

Are the following functions identical
f (x) = logx² and Q (x) = 2logx

a) Identical

b) Not identical

c) Cannot say

Are the following functions identical
 

a) Identical

b) Not identical

c) Cannot say

Are the following functions identical
f (x) = 1 and  Q (x) = sin²x + cos²x

a) Identical

b) Not identical

c) Cannot say

In what interval are the following functions identical
f(x) = x and g(x) =

a) (−∞, ∞)

b) (−∞, 0)

c) (0, ∞)

d) [0, ∞)

In what interval are the following functions identical
 

a) (−∞, ∞)

b) [0, ∞)

c) [1, ∞)

d) (1, ∞)

Solve the inequality
 

a) x > 5

b) x ≥ 5

c) x = 5

d) −∞ < x ≤ 5

Prove that the inequality
 
has no solution.

Prove that if for a linear function f (x) = ax + b, the value of the argument x = (n = 1, 2, 3.  .  .) form an arithmetic progression then the corresponding value of  the function   = f () (n = 1, 2, 3, ...) also form an arithmetic progression.

Prove that the function f (x) defined in a symmetrical interval (−l, l) can be represented as a sum of an even function and an odd function.

Rewrite the following function in the form of a sum of an even and an odd function
 

Prove that if the function f(x) = sinx + cosax is periodic, then a is a rational number.

Test the following function for monotony
f (x) = |x|

a) Monotonically increasing everywhere

b) Monotonically decreasing everywhere

c) Monotonically increasing for x ≥ 0 and monotonically decreasing for x < 0

d) None of the above