Advertisements
Advertisements
प्रश्न
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
उत्तर
f : Z → Z, given by f(x) = x3
Injection test:
Let x and y be any two elements in the domain (Z), such that f(x) = f(y)
f(x) = f(y)
x3 = y3
x = y
So, f is an injection.
Surjection test :
Let y be any element in the co-domain (Z), such that f(x) = y for some element x in Z(domain).
f(x) = y
x3 = y
x = 3sqrty which may not be in Z.
For example, if y = 3,
x =`3 sqrt3` is not in Z .
So, f is not a surjection and f is not a bijection .
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f: Z → Z given by f(x) = x3
Let f: N → N be defined by f(n) = `{((n+1)/2, ",if n is odd"),(n/2,",n is even"):}` for all n ∈ N.
State whether the function f is bijective. Justify your answer.
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
Let A = {−1, 0, 1} and f = {(x, x2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 8x3 and g(x) = x1/3.
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) = ex.
Give examples of two functions f : N → N and g : N → N, such that gof is onto but f is not onto.
If f : A → B and g : B → C are onto functions, show that gof is a onto function.
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
If f(x) = |x|, prove that fof = f.
Let f, g, h be real functions given by f(x) = sin x, g (x) = 2x and h (x) = cos x. Prove that fog = go (fh).
Show that the function f : Q → Q, defined by f(x) = 3x + 5, is invertible. Also, find f−1
If f : Q → Q, g : Q → Q are two functions defined by f(x) = 2 x and g(x) = x + 2, show that f and g are bijective maps. Verify that (gof)−1 = f−1 og −1.
If f : A → A, g : A → A are two bijections, then prove that fog is an injection ?
If A = {1, 2, 3} and B = {a, b}, write the total number of functions from A to B.
If A = {a, b, c} and B = {−2, −1, 0, 1, 2}, write the total number of one-one functions from A to B.
If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
If f : C → C is defined by f(x) = x4, write f−1 (1).
Let f : R − {−1} → R − {1} be given by\[f\left( x \right) = \frac{x}{x + 1} . \text{Write } f^{- 1} \left( x \right)\]
Let `f : R - {- 3/5}` → R be a function defined as `f (x) = (2x)/(5x +3).`
f-1 : Range of f → `R -{-3/5}`.
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
Let f : R → R be the function defined by f(x) = 4x − 3 for all x ∈ R Then write f . [NCERT EXEMPLAR]
Let
f : R → R be given by
\[f\left( x \right) = \left[ x^2 \right] + \left[ x + 1 \right] - 3\]
where [x] denotes the greatest integer less than or equal to x. Then, f(x) is
(d) one-one and onto
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 ) :}`
If \[f : R \to R\] is given by \[f\left( x \right) = x^3 + 3, \text{then} f^{- 1} \left( x \right)\] is equal to
Let \[f\left(x\right) = x^3\] be a function with domain {0, 1, 2, 3}. Then domain of \[f^{-1}\] is ______.
Mark the correct alternative in the following question:
If the set A contains 7 elements and the set B contains 10 elements, then the number one-one functions from A to B is
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, 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 R be the set of real numbers and f: R → R be the function defined by f(x) = 4x + 5. Show that f is invertible and find f–1.
Let f: R → R be defined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are ______.
Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______
Let f: R – `{3/5}` → R be defined by f(x) = `(3x + 2)/(5x - 3)`. Then ______.
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, is ____________.
Let f: R → R defined by f(x) = 3x. Choose the correct answer
Let n(A) = 4 and n(B) = 6, Then the number of one – one functions from 'A' to 'B' is:
Let a and b are two positive integers such that b ≠ 1. Let g(a, b) = Number of lattice points inside the quadrilateral formed by lines x = 0, y = 0, x = b and y = a. f(a, b) = `[a/b] + [(2a)/b] + ... + [((b - 1)a)/b]`, then the value of `[(g(101, 37))/(f(101, 37))]` is ______.
(Note P(x, y) is lattice point if x, y ∈ I)
(where [.] denotes greatest integer function)
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 ______.
