Advertisements
Advertisements
प्रश्न
Give an example of a function which is one-one but not onto ?
Advertisements
उत्तर
which is one-one but not onto.
f: Z → Z given by f(x) = 3x + 2
Injectivity:
Let x and y be any two elements in the domain (Z), such that f(x) = f(y).
f (x)= f(y)
⇒ 3x + 2 =3y + 2
⇒ 3x = 3y
⇒ x = y
⇒ f(x) = f(y) ⇒ x = y
So, f is one-one.
Surjectivity:
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
⇒ 3x + 2 = y
⇒ 3x = y - 2
⇒ x= `(y - 2)/3`. It may not be in the domain (Z)
because if we take y = 3,
`x = (y - 2)/3 = (3-2)/3 = 1/3 ∉` domain Z.
So, for every element in the co domain there need not be any element in the domain such that f(x) = y.
Thus, f is not onto.
APPEARS IN
संबंधित प्रश्न
Show that the function f : R* → R* defined by f(x) = `1/x` is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R?
Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but gis not injective.
(Hint: Consider f(x) = x and g(x) =|x|)
Let S = {a, b, c} and T = {1, 2, 3}. Find F−1 of the following functions F from S to T, if it exists.
F = {(a, 2), (b, 1), (c, 1)}
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.)
Let A = {−1, 0, 1} and f = {(x, x2) : x ∈ A}. Show that f : A → A 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) = |x|
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x2 + x
If A = {1, 2, 3}, show that a one-one function f : A → A must be onto.
Suppose f1 and f2 are non-zero one-one functions from R to R. Is `f_1 / f^2` necessarily one - one? Justify your answer. Here,`f_1/f_2 : R → R is given by (f_1/f_2) (x) = (f_1(x))/(f_2 (x)) for all x in R .`
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) = 8x3 and g(x) = x1/3.
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
Find fog and gof if : f (x) = x+1, g(x) = `e^x`
.
` if f : (-π/2 , π/2)` → R and g : [−1, 1]→ R be defined as f(x) = tan x and g(x) = `sqrt(1 - x^2)` respectively, describe fog and gof.
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 = {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.
Which one the following relations on A = {1, 2, 3} is a function?
f = {(1, 3), (2, 3), (3, 2)}, g = {(1, 2), (1, 3), (3, 1)} [NCERT EXEMPLAR]
Let A = {a, b, c, d} and f : A → A be given by f = {( a,b ),( b , d ),( c , a ) , ( d , c )} write `f^-1`. [NCERT EXEMPLAR]
If a function\[f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5\] is a bijection, then B =
\[f : Z \to Z\] be given by
` f (x) = {(x/2, ", if x is even" ) ,(0 , ", if x is odd "):}`
Then, f is
Let \[f\left( x \right) = x^2 and g\left( x \right) = 2^x\] Then, the solution set of the equation
Let [x] denote the greatest integer less than or equal to x. If \[f\left( x \right) = \sin^{- 1} x, g\left( x \right) = \left[ x^2 \right]\text{ and } h\left( x \right) = 2x, \frac{1}{2} \leq x \leq \frac{1}{\sqrt{2}}\]
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.
Let A be a finite set. Then, each injective function from A into itself is not surjective.
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.
Let f: R → R be given by f(x) = tan x. Then f–1(1) is ______.
Let f : R → R, g : R → R be two functions such that f(x) = 2x – 3, g(x) = x3 + 5. The function (fog)-1 (x) is equal to ____________.
Let R be a relation on the set L of lines defined by l1 R l2 if l1 is perpendicular to l2, then relation R is ____________.
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Let R: B → G be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then R is ____________.
If f: R → R given by f(x) =(3 − x3)1/3, find f0f(x)
A function f: x → y is said to be one – one (or injective) if:
If f; R → R f(x) = 10x + 3 then f–1(x) is:
Number of integral values of x satisfying the inequality `(3/4)^(6x + 10 - x^2) < 27/64` is ______.
Let f(n) = `[1/3 + (3n)/100]n`, where [n] denotes the greatest integer less than or equal to n. Then `sum_(n = 1)^56f(n)` is equal to ______.
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.
Find the domain of sin–1 (x2 – 4).
