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.
My Self Assessment
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
Prove the identity
Find the domain of definition of the function
a) {0}
b) {0} ∪ [1, ∞)
c) (0, ∞)
d) [0, ∞)
a) x ≥ 0
b) 0 ≤ x ≤ 2nπ
c) 0 ≤ x ≤ (2nπ)2
d) (2nπ)2 ≤ x ≤((2n + 1)π)2, n = 0, 1, 2. . .
a) x = 0
b) x = 0, 1, 2. . .
c) x = 0, ±1, ±2, . . .
d) Defined for all x except x = 0, ±1, ±2, . . .
b) (0, ∞)
c) (−∞, 0)
d) (−∞, ∞)
d) Not defined for any x
a) [−4, −2]
b) [−4, 2]
c) [−4, 2] ∪ {4}
d) [−4, −2] ∪ [2, 4]
a) x = 0, π
b) x = nπ
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
Are the following functions identical f (x) = 1 and Q (x) = sin2x + cos2x
In what interval are the following functions identical f(x) = x and g(x) =
a) (−∞, ∞)
b) (−∞, 0)
In what interval are the following functions identical
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 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
