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Relations And Functions

Question

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The set of points where x f(x) = x /1+|x| is differentiable is

(−∞, 0) ∪ (0, ∞)

(−∞, −1) ∪ (−1, ∞)

(−∞, ∞)

(0, ∞)

Solution

C.

(−∞, ∞)

open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator 1 minus straight x end fraction comma space straight x space less than 0 end cell row cell fraction numerator straight x over denominator 1 plus straight x end fraction comma space straight x space greater or equal than 0 end cell end table close rightwards double arrow space straight f apostrophe left parenthesis straight x right parenthesis space equals space open curly brackets table attributes columnalign left end attributes row cell straight x over open parentheses 1 minus straight x close parentheses squared comma space straight x space less than 0 end cell row cell straight x over open parentheses 1 plus straight x close parentheses squared comma space straight x space greater or equal than 0 end cell end table close

∴ f′(x) exist at everywhere.

Some More Questions From Relations and Functions Chapter

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L₁, L₂) : L₁ is perpendicular to L₂}. Show that R is symmetric but neither reflexive nor transitive.

Determine whether each of the following relations are reflexive, symmetric and transitive :

(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as

R = {(x, y) : 3 x – y = 0}

(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x,y) : x – y is an integer}

(v) Relation R in the set A of human beings in a town at a particular time given by
(a)    R = {(x, y) : x and y work at the same place}
(b)    R = {(x,y) : x and y live in the same locality}
(c)    R = {(x, y) : x is exactly 7 cm taller than y}
(d)    R = {(x, y) : x is wife of y}
(e)    R = {(x,y) : x is father of y}

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b³} is refleive, symmetric or transitive.

If R and R’ arc reflexive relations on a set then so are R ∪ R’ and R ∩ R’.

If R and R’ are symmetric relations on a set A, then R ∩ R’ is also a sysmetric relation on A.

Show that the union of two symmetric relations on a set is again a symmetric relation on that set.

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