Let f : X → Y be an invertibl | 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.

Find the area of the region bounded by the curve y² = x and the lines x = 1, x = 4 and the x-axis.

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

 Consider f : {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f ^–1 and show that (f ^–1)^–1 = f. 

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

Letbe a function defined as. The inverse of f is map g: Range

(A)  

(B) 

(C)  

(D)

In Figure, identify the following vectors.

(i) Coinitial (ii) Equal (iii) Collinear but not equal

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

is neither one-one nor onto.

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)}