Advertisements
Advertisements
प्रश्न
Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.
Advertisements
उत्तर
We know that every onto function from A to itself is one-one.
So, the number of one-one functions = number of bijections = n!
APPEARS IN
संबंधित प्रश्न
Show that the function f in `A=R-{2/3} ` defined as `f(x)=(4x+3)/(6x-4)` is one-one and onto hence find f-1
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.
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 1 + x2
Show that the function f : R → R given by f(x) = x3 is injective.
Given examples of two functions f: N → N and g: N → N such that gof is onto but f is not onto.
(Hint: Consider f(x) = x + 1 and `g(x) = {(x-1, ifx >1),(1, if x = 1):}`
Find the number of all onto functions from the set {1, 2, 3, ..., n} to itself.
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]?
Which of the following functions from A to B are one-one and onto ?
f3 = {(a, x), (b, x), (c, z), (d, z)} ; A = {a, b, c, d,}, B = {x, y, z}.
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + 3 and g(x) = x2 + 5 .
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) = `x^2` + 2 , g (x) = 1 − `1/ (1-x)`.
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
Consider f : R+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with `f^-1 (x) = (sqrt (x +6)-1)/3 .`
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]
Let A = {x &epsis; R | −1 ≤ x ≤ 1} and let f : A → A, g : A → A be two functions defined by f(x) = x2 and g(x) = sin (π x/2). Show that g−1 exists but f−1 does not exist. Also, find g−1.
Let f be a function from R to R, such that f(x) = cos (x + 2). Is f invertible? Justify your answer.
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).
Let f : R − {−1} → R − {1} be given by\[f\left( x \right) = \frac{x}{x + 1} . \text{Write } f^{- 1} \left( x \right)\]
Write the domain of the real function
`f (x) = sqrt([x] - x) .`
\[f : R \to R\] is defined by
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is\]
The function
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 \[-\] \[\left\{ \frac{3}{5} \right\}\] \[\to\] R be defined by f(x) = \[\frac{3x + 2}{5x - 3}\] Then,
Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.
Let A = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = `(x - 2)/(x - 3)` ∀ x ∈ A . Then show that f is bijective.
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
f(x) = `x/2`
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
The function f : R → R given by f(x) = x3 – 1 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 ____________.
If f: R → R given by f(x) =(3 − x3)1/3, find f0f(x)
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.
Which one of the following graphs is a function of x?
![]() |
![]() |
| Graph A | Graph B |


