define signum function, greatest integer function, modulus function and provide their graphs with explanation.
Answers
this is the formula
Step-by-step explanation:
RThis is known as signum function.Let us check value of f(x) for different values of xFor x = –1x < 0So, f(x) ...
EXPLANATION.
(1) = Signum Function.
Signum function is defined by sgn(x) defined as,
f(x) = sgn(x) = 1, x > 0.
0, x = 0. = |x|/x, x ≠ 0.
-1, x < 0 0, x = 0.
Domain of the function is R.
Range of the function is {-1,0,1}.
We can also defined sgn[f(x)] as,
sgn[f(x)] = 1, f(x) > 0
0, f(x) = 0. = |f(x)|/f(x), f(x) ≠ 0.
-1, f(x) < 0. 0, f(x) = 0.
(2) = Greatest integer function.
A function is said to be greatest integer function if it is of the form of f(x) = [x] where [x] = integer equal or less than x.
Properties of G.I.F.
(1) = [x] = x if x is integer.
(2) = [x + I] = [x] + I, if I is an integer.
(3) = [x + y] ≥ [x] + [y].
(4) = If [Ф(x)] ≥ I then Ф(x) ≥ I.
(5) = If [Ф(x)] ≤ I then Ф(x) < I + 1.
(6) = If [x] > n ⇒ x ≥ n + 1.
(7) = If [x] < n ⇒ x < n, n ∈ I.
(8) = [-x] = -[x] if ∨ x ∈ I.
(9) = [-x] = -[x] - 1 if x ∉ integer.
(10) = [x + y] = [x] + [y + x - [x]] ∨ x, y ∈ R.
(11) = [x] + [x + 1/n] + [x + 2/n] + ,,,,,+[x + n-1/n] = [nx].
(3) = Modulus Functions.
It is given n ∈ N by y = |x| = x, x ≥ 0.
-x, x < 0.
Properties of modulus function.
(1) = |x| ≤ a ⇒ -a ≤ x ≤ a.
(2) = |x| ≥ a ⇒ x ≤ -a Or x ≥ a.
(3) = |x + y| = |x| + |y| ⇒ x, y ≥ 0 Or x ≤ 0, y ≤ 0.
(4) = |x - y| = |x| - |y| ⇒ x ≥ 0 and |x| ≥ |y| Or x ≤ 0 and y ≤ 0 and |x| ≥ |y|.
(5) = |x ± y| ≤ |x| + |y|.
(6) = |x ± y| ≥ |x| - |y|.
