PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Set of ordered pair of a function ? If so, examine whether the mapping is injective or surjective :{(a, b) : a is a person, b is an ancestor of a} 

Let A = {1, 2, 3}. Write all one-one from A to itself.

If f : R → R be the function defined by f(x) = 4x³ + 7, show that f is a bijection.

Show that the exponential function f : R → R, given by f(x) = e^x, is one-one but not onto. What happens if the co-domain is replaced by`R0^+` (set of all positive real numbers)?

Show that the logarithmic function  f : R0⁺ → R   given  by f (x)  log_a x ,a> 0   is   a  bijection.

If A = {1, 2, 3}, show that a one-one function f : A → A must be onto.

If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.

Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.

Give examples of two one-one functions f₁ and f₂ from R to R, such that f₁ + f₂ : R → R. defined by (f₁ + f₂) (x) = f₁ (x) + f₂ (x) is not one-one.

Give examples of two surjective functions f₁ and f₂ from Z to Z such that f₁ + f₂ is not surjective.

Show that if f₁ and f₂ are one-one maps from R to R, then the product f₁ × f₂ : R → R defined by (f₁ × f₂) (x) = f₁ (x) f₂ (x) need not be one - one.

Suppose f₁ and f₂ are non-zero one-one functions from R to R. Is `f_1 / f^2` necessarily one - one? Justify your answer. Here,`f_1/f_2 : R → R   is   given   by   (f_1/f_2) (x) = (f_1(x))/(f_2 (x))  for all  x in R .`

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:
(i) an injective map from A to B
(ii) a mapping from A to B which is not injective
(iii) a mapping from A to B.

Show that f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.

Let f : N → N be defined by

`f(n) = { (n+ 1, if n  is  odd),( n-1 , if n  is  even):}`

Show that f is a bijection. 

                      [CBSE 2012, NCERT]

Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + 3 and  g(x) = x² + 5 .

Find gof and fog when f : R → R and g : R → R is defined by  f(x) = 2x + x² and  g(x) = x³

Find gof and fog when f : R → R and g : R → R is defined by  f(x) = x² + 8 and g(x) = 3x³ + 1 .

Find gof and fog when f : R → R and g : R → R is defined by  f(x) = x and g(x) = |x| .

Find gof and fog when f : R → R and g : R → R is defined by  f(x) = x² + 2x − 3 and  g(x) = 3x − 4 .