Advertisements
Advertisements
प्रश्न
Which of the following functions form Z to itself are bijections?
पर्याय
\[f\left( x \right) = x^3\]
\[f\left( x \right) = x + 2\]
\[f\left( x \right) = 2x + 1\]
\[f\left( x \right) = x^2 + x\]
Advertisements
उत्तर
f is not onto because for y = 3∈Co-domain(Z), there is no value of x∈Domain(Z)
\[ x^3 = 3\]
\[ \Rightarrow x = \sqrt[3]{3} \not\in Z\]
⇒ f is not onto
So, f is not a bijection
(b) Injectivity:
Let x and y be two elements of the domain (Z), such that
\[x + 2 = y + 2\]
\[ \Rightarrow x = y\]
Surjectivity:
Let y be an element in the co-domain (Z), such that
\[ \Rightarrow y = x + 2\]
\[ \Rightarrow x = y - 2 \in Z \left( Domain \right)\]
So, f is a bijection.
\[ \Rightarrow 4 = 2x + 1\]
\[ \Rightarrow 2x = 3\]
\[ \Rightarrow x = \frac{3}{2} \not\in Z\]
So,f is not a bijection.
\[\]
\[\left( d \right) f\left( 0 \right) = 0^2 + 0 = 0\]
\[and f\left( - 1 \right) = \left( - 1 \right)^2 + \left( - 1 \right) = 1 - 1 = 0\]
⇒ 0 and -1 have the same image.
⇒ f is not one-one.
So,fis not a bijection.
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 function f : N → N, defined by f(x) = x2 + x + 1, is one-one but not onto
If f : A → B is an injection, such that range of f = {a}, determine the number of elements in A.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + 3 and g(x) = x2 + 5 .
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x2 + 2x − 3 and g(x) = 3x − 4 .
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.
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
If f : A → B and g : B → C are onto functions, show that gof is a onto function.
Let A = {1, 2, 3, 4}; B = {3, 5, 7, 9}; C = {7, 23, 47, 79} and f : A → B, g : B → C be defined as f(x) = 2x + 1 and g(x) = x2 − 2. Express (gof)−1 and f−1 og−1 as the sets of ordered pairs and verify that (gof)−1 = f−1 og−1.
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
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).
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)}.
Which of the following graphs represents a one-one function?
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).
If f : C → C is defined by f(x) = (x − 2)3, write f−1 (−1).
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 A = {1, 2, 3, 4} and B = {a, b} be two sets. Write the total number of onto functions from A to B.
If f : R → R be defined by f(x) = (3 − x3)1/3, then find fof (x).
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.
If f(x) = 4 −( x - 7)3 then write f-1 (x).
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
The function f : R → R defined by
`f (x) = 2^x + 2^(|x|)` is
Let
\[f : R \to R\] be a function defined by
Let
\[f : R - \left\{ n \right\} \to R\]
A function f from the set of natural numbers to the set of integers defined by
\[f\left( n \right)\begin{cases}\frac{n - 1}{2}, & \text{when n is odd} \\ - \frac{n}{2}, & \text{when n is even}\end{cases}\]
The function \[f : R \to R\] defined by
\[f\left( x \right) = 6^x + 6^{|x|}\] is
If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\]
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f−1.
Write about strlen() function.
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 ______.
Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______
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.
Range of `"f"("x") = sqrt((1 - "cos x") sqrt ((1 - "cos x")sqrt ((1 - "cos x")....infty))`
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 function of degree 6 such that `d/dx (f(x))` = (x – 1)3 (x – 3)2, then
Assertion (A): f(x) has a minimum at x = 1.
Reason (R): When `d/dx (f(x)) < 0, ∀ x ∈ (a - h, a)` and `d/dx (f(x)) > 0, ∀ x ∈ (a, a + h)`; where 'h' is an infinitesimally small positive quantity, then f(x) has a minimum at x = a, provided f(x) is continuous at x = a.
