Advertisements
Advertisements
प्रश्न
Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) = cat. Show that f, g and gof are invertible. Find f−1, g−1 and gof−1and show that (gof)−1 = f −1o g−1
उत्तर
f = { ( 1, a ) . (2, b) , (c , 3 ) } and g = {(a , apple) , (b , ball) , (c , cat)} Clearly , f and g are bijections.
So, f and g are invertible.
Now,
f -1 = {(a ,1) , (b , 2) , (3,c)} and g-1 = {(apple, a ) , (ball ,b), (cat , c)}
So, f-1 o g-1= {apple , 1} , (ball,2), (cat , 3 )} ......... (1)
f : {1,2,3,} → {a,b,c} and g : {a,b,c} → {apple , ball , cat}
So, gof : {1.2.3} → {apple , ball, cat}
⇒ (gof) (1) =g (f(1)) = g (a) = apple
(gof) (2) = g (f (2)) = g (b) = ball,
and (gof) (3) = g (f(3)) = g (c) cat
∴ gof = {(1 . apple) ,(2, ball) , (3 , cat)}
Clearly , gof is a bijection.
So, gof is invertible.
(gof)-1 = {(apple , 1), (ball,2),(cat , 3)} ....... (2)
Form (1) and (2) , we get :
(gof)-1 = f-1 o g -1
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f: N → N given by f(x) = x2
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.
Let f: R → R be the Signum Function defined as
f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`
and g: R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0, 1]?
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 + 1
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sin2x + cos2x
Classify the following function as injection, surjection or bijection :
f : Q − {3} → Q, defined by `f (x) = (2x +3)/(x-3)`
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}
If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.
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.
Find fog and gof if : f (x) = x2 g(x) = cos x .
Find fog and gof if : f(x) = c, c ∈ R, g(x) = sin `x^2`
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.
Find f −1 if it exists : f : A → B, where A = {1, 3, 5, 7, 9}; B = {0, 1, 9, 25, 49, 81} and f(x) = x2
Consider the function f : R+ → [-9 , ∞ ]given by f(x) = 5x2 + 6x - 9. Prove that f is invertible with f -1 (y) = `(sqrt(54 + 5y) -3)/5` [CBSE 2015]
Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.
If f : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).
If f : R → R, g : R → are given by f(x) = (x + 1)2 and g(x) = x2 + 1, then write the value of fog (−3).
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
If f : R → R be defined by f(x) = (3 − x3)1/3, then find fof (x).
If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = \[\alpha x + \beta\] then find the values of \[\alpha\] and \[ \beta\] . [NCERT EXEMPLAR]
If f(x) = 4 −( x - 7)3 then write f-1 (x).
The function
f : A → B defined by
f (x) = - x2 + 6x - 8 is a bijection if
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
Let
\[f : R \to R\] is defined by
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is\]
\[f : Z \to Z\] be given by
` f (x) = {(x/2, ", if x is even" ) ,(0 , ", if x is odd "):}`
Then, f is
Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f
Let N be the set of natural numbers and the function f: N → N be defined by f(n) = 2n + 3 ∀ n ∈ N. Then f is ______.
Let f: R → R be defined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are ______.
For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is surjective. Then g is surjective.
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}
Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
f = {(1, 4), (1, 5), (2, 4), (3, 5)}
Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.
Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.
Answer the following questions using the above information.
- The function f: Z → Z defined by f(x) = x2 is ____________.
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.
If A = {x ∈ R: |x – 2| > 1}, B = `{x ∈ R : sqrt(x^2 - 3) > 1}`, C = {x ∈ R : |x – 4| ≥ 2} and Z is the set of all integers, then the number of subsets of the set (A ∩ B ∩ C) C ∩ Z is ______.
The given function f : R → R is not ‘onto’ function. Give reason.
