हिंदी

State with Reasons Whether the Following Functions Have Inverse : G : {5, 6, 7, 8} → {1, 2, 3, 4} With G = {(5, 4), (6, 3), (7, 4), (8, 2)}

Advertisements
Advertisements

प्रश्न

State with reason whether the following functions have inverse :

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

Advertisements

उत्तर

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

⇒ f is not one-one.

⇒ f is not a bijection.
So, f does not have an inverse.

shaalaa.com
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 2: Functions - Exercise 2.4 [पृष्ठ ६८]

APPEARS IN

आर.डी. शर्मा Mathematics Volume 1 and 2 [English] Class 12
अध्याय 2 Functions
Exercise 2.4 | Q 1.2 | पृष्ठ ६८

वीडियो ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्न

Let f : N → N be defined by f(n) = `{((n+1)/2", if n is odd"),(n/2", if n is even"):}` for all n ∈ N.

State whether the function f is bijective. Justify your answer.


Let A = {–1, 0, 1, 2}, B = {–4, –2, 0, 2} and f, g : A → B be functions defined by f(x) = x2 – x, x ∈ A and g(x) = `2|x - 1/2| – 1`, x ∈ A. Are f and g equal?

Justify your answer. (Hint: One may note that two functions f : A → B and g : A → B such that f(a) = g(a) ∀ a ∈ A are called equal functions.)


Which of the following functions from A to B are one-one and onto?

 f2 = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {abc}


Prove that the function f : N → N, defined by f(x) = x2 + x + 1, is one-one but not onto


Let A = {−1, 0, 1} and f = {(xx2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.


Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = |x|


Classify the following function as injection, surjection or bijection :

f : Q − {3} → Q, defined by `f (x) = (2x +3)/(x-3)`


Classify the following function as injection, surjection or bijection :

f : Q → Q, defined by f(x) = x3 + 1


Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 5x3 + 4


Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : g(x) = |x|  


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


Find gof and fog when f : R → R and g : R → R is  defined by  f(x) = 8x3 and  g(x) = x1/3.


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.


 Find fog and gof  if  : f (x) = ex g(x) = loge x .


Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.


Let f be a real function given by f (x)=`sqrt (x-2)`
Find each of the following:

(i) fof
(ii) fofof
(iii) (fofof) (38)
(iv) f2

Also, show that fof ≠ `f^2` .


Consider f : R → R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with inverse f−1 of f given by f−1 `(x)= sqrt (x-4)` where R+ is the set of all non-negative real numbers.


Let f : R `{- 4/3} `- 43 →">→ R be a function defined as f(x) = `(4x)/(3x +4)` . Show that f : R - `{-4/3}`→ Rang (f) is one-one and onto. Hence, find f -1.


Let f : [−1, ∞) → [−1, ∞) be given by f(x) = (x + 1)2 − 1, x ≥ −1. Show that f is invertible. Also, find the set S = {x : f(x) = f−1 (x)}.


Let f be a function from R to R, such that f(x) = cos (x + 2). Is f invertible? Justify your answer.


Which of the following graphs represents a one-one function?


Let `f : R - {- 3/5}` → R be a function defined as `f  (x) = (2x)/(5x +3).` 

f-1 : Range of f → `R -{-3/5}`.


A function f  from the set of natural numbers to integers defined by

`{([n-1]/2," when  n is  odd"   is ),(-n/2,when  n  is  even ) :}`

 

 


The function

\[f : R \to R, f\left( x \right) = x^2\]
 

A function f: R→ R defined by f(x) = `(3x) /5 + 2`, x ∈ R. Show that f is one-one and onto. Hence find f−1.


Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f–1 


If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one-one from A onto A. Find f–1


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 the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.


Let f: R → R be given by f(x) = tan x. Then f–1(1) is ______.


Let A = R – {3}, B = R – {1}. Let f : A → B be defined by `"f"("x") = ("x" - 2)/("x" - 3)` Then, ____________.


Let f : [0, ∞) → [0, 2] be defined by `"f" ("x") = (2"x")/(1 + "x"),` then f is ____________.


Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.

Answer the following using the above information.

  • Let f: R → R be defined by f(x) = x − 4. Then the range of f(x) 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.

  • Let : N → R be defined by f(x) = x2. Range of the function among the following is ____________.

The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.


Let [x] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)` is (–∞, a) ∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is ______.


Let f(x) be a polynomial of degree 3 such that f(k) = `-2/k` for k = 2, 3, 4, 5. Then the value of 52 – 10f(10) is equal to ______.


If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×