Advertisements
Advertisements
प्रश्न
If A = {1, 2, 3, 4} and B = {a, b, c, d}, define any four bijections from A to B. Also give their inverse functions.
Advertisements
उत्तर
f1={ (1, a), (2, b), (3, c), (4, d)} ⇒ f1−1 ={(a, 1), (b, 2), (c, 3), (d, 4)}
f2={(1, b), (2, a), (3, c), (4, d)} ⇒f2−1 {(b, 1), (a, 2), (c, 3), (d, 4)}
f3= { (1, a), (2, b), (4, c), (3, d)} ⇒ f3−1 {(a, 1), (b, 2), (c, 4), (d, 3)}
f4 = {(1, b), (2, a), (4, c), (3, d)} ⇒ f4−1 = {(b, 1), (a, 2), (c, 4), (d, 3)}
Clearly, all these are bijections because they are one-one and onto.
APPEARS IN
संबंधित प्रश्न
Show that the function f : R → R given by f(x) = x3 is injective.
Give an example of a function which is neither one-one nor onto ?
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 1 + x2
Find fog and gof if : f (x) = x+1, g(x) = `e^x`
.
Find fog and gof if : f(x) = c, c ∈ R, g(x) = sin `x^2`
If f : R → R be defined by f(x) = x3 −3, then prove that f−1 exists and find a formula for f−1. Hence, find f−1(24) and f−1 (5).
A function f : R → R is defined as f(x) = x3 + 4. Is it a bijection or not? In case it is a bijection, find f−1 (3).
Let A = R - {3} and B = R - {1}. Consider the function f : A → B defined by f(x) = `(x-2)/(x-3).`Show that f is one-one and onto and hence find f-1.
[CBSE 2012, 2014]
If f : C → C is defined by f(x) = (x − 2)3, write f−1 (−1).
If the mapping f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3}, given by f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, then write fog. [NCERT EXEMPLAR]
Let the function
\[f : R - \left\{ - b \right\} \to R - \left\{ 1 \right\}\]
\[f\left( x \right) = \frac{x + a}{x + b}, a \neq b .\text{Then},\]
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 ) :}`
Which of the following functions from
to itself are bijections?
If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\]
If \[f : R \to \left( - 1, 1 \right)\] is defined by
\[f\left( x \right) = \frac{- x|x|}{1 + x^2}, \text{ then } f^{- 1} \left( x \right)\] equals
Mark the correct alternative in the following question:
Let f : R→ R be defined as, f(x) = \[\begin{cases}2x, if x > 3 \\ x^2 , if 1 < x \leq 3 \\ 3x, if x \leq 1\end{cases}\]
Then, find f( \[-\]1) + f(2) + f(4)
Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is ______.
Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.
Let f: R – `{3/5}` → R be defined by f(x) = `(3x + 2)/(5x - 3)`. Then ______.
Let f: `[2, oo)` → R be the function defined by f(x) = x2 – 4x + 5, then the range of f is ______.
If f(x) = (4 – (x – 7)3}, then f–1(x) = ______.
Let A = {0, 1} and N be the set of natural numbers. Then the mapping f: N → A defined by f(2n – 1) = 0, f(2n) = 1, ∀ n ∈ N, is onto.
Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, is ____________.
Let f : [0, ∞) → [0, 2] be defined by `"f" ("x") = (2"x")/(1 + "x"),` then f is ____________.
Let f: R → R defined by f(x) = 3x. Choose the correct answer
A function f: x → y is said to be one – one (or injective) if:
Prove that the function f is surjective, where f: N → N such that `f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if "n is even"):}` Is the function injective? Justify your answer.
Let f(1, 3) `rightarrow` R be a function defined by f(x) = `(x[x])/(1 + x^2)`, where [x] denotes the greatest integer ≤ x, Then the range of f is ______.
The graph of the function y = f(x) is symmetrical about the line x = 2, then ______.
Let a function `f: N rightarrow N` be defined by
f(n) = `{:[(2n",", n = 2"," 4"," 6"," 8","......),(n - 1",", n = 3"," 7"," 11"," 15","......),((n + 1)/2",", n = 1"," 5"," 9"," 13","......):}`
then f 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 ______.
A function is a rule that assigns each element of one set to exactly how many elements of another set?
A function is called many-one if two or more different elements of the domain have:
In the function from \(X\) to \(Y\), if \(6\in Y\) is not the image of any element of \(X\), the function is:
Which real-life situation resembles a one-one idea when no roll number is repeated?
