If f: R → R be given by f(x) | Class 12 Mathematics Chapter Relations and Functions, Relations and Functions NCERT Solutions

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i) f : R → R defined by f(x) = 3 – 4x

(ii) f : R → R defined by f(x) = 1 + x² 

 Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

 Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Answer the following as true or false.

 \begin{align}(i) \overrightarrow{a}\;  and\; \overrightarrow{-a}\; are\; collinear.\end{align}

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

Show that the Signum Function f : R → R, given by

is neither one-one nor onto.

The rate of change of the area of a circle with respect to its radius r at r = 6 cm is

(A) 10π (B) 12π (C) 8π (D) 11π

 Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.

 Let f : X → Y be an invertible function. Show that f has unique inverse.

(Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1(y) = 1Y(y) = fog2(y). Use one-one ness of f). 

Let f, g and h be functions from R to R. Show that

(f + g)oh = foh + goh

(f . g)oh = (foh) . (goh) 

State with reason whether following functions have inverse

(i) f : {1, 2, 3, 4} → {10} with

f  = {(1, 10), (2, 10), (3, 10), (4, 10)}

(ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with

g = {(5, 4), (6, 3), (7, 4), (8, 2)}

(iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with

h = {(2, 7), (3, 9), (4, 11), (5, 13)}