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

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 .

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

Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.

Let f = {(1, −1), (4, −2), (9, −3), (16, 4)} and g = {(−1, −2), (−2, −4), (−3, −6), (4, 8)}. Show that gof is defined while fog is not defined. Also, find gof.

Let A = {a, b, c}, B = {u v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as :
f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.
Show that f and g both are bijections and find fog and gof.

Find  fog (2) and gof (1) when : f : R → R ; f(x) = x² + 8 and g : R → R; g(x) = 3x³ + 1.

Let R⁺ be the set of all non-negative real numbers. If f : R⁺ → R⁺ and g : R⁺ → R⁺ are defined as `f(x)=x^2` and `g(x)=+sqrtx` , find fog and gof. Are they equal functions ?

Let f : R → R and g : R → R be defined by f(x) = x² and g(x) = x + 1. Show that fog ≠ gof.

Let f : R → R and g : R → R be defined by f(x) = x + 1 and g (x) = x − 1. Show that fog = gof = I_R.

Verify associativity for the following three mappings : f : N → Z₀ (the set of non-zero integers), g : Z₀ → Q and h : Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = e^x.