Advertisements
Advertisements
प्रश्न
\[f : Z \to Z\] be given by
` f (x) = {(x/2, ", if x is even" ) ,(0 , ", if x is odd "):}`
Then, f is
पर्याय
onto but not one-one
one-one but not onto
one-one and onto
neither one-one nor onto
Advertisements
उत्तर
Injectivity:
Let x and y be two elements in the domain (Z), such that
LaTeX
\[f\left( x \right) = f\left( y \right)\]
\[Case-1: \text{Let both x andybe even}.\]
\[\text{Then},\]
\[f\left( x \right) = f\left( y \right)\]
\[ \Rightarrow \frac{x}{2} = \frac{y}{2}\]
\[ \Rightarrow x = y\]
\[Case-2: \text{Let bothx andybe odd}.\]
\[\text{Then},\]
\[f\left( x \right) = f\left( y \right)\]
\[ \Rightarrow 0 = 0\]
\[\text{Here, we cannot determine whether } x = y.\] So, f is not one-one.
Surjectivity:
Let y be an element in the co-domain (Z), such that
\[\text{Co-domain of f} = Z = \left\{ 0, \pm 1, \pm 2, \pm 3, \pm 4, . . . \right\} \]
\[\text{Range of f} = \left\{ 0, 0, \frac{\pm 2}{2}, 0, \frac{\pm 4}{2} , . . . \right\} = \left\{ 0, \pm 1, \pm 2, . . . \right\} \]
\[ \Rightarrow \text{Co-domain of f} = \text{Range of f}\]
\[\Rightarrow\] f is onto.
So, the answer is (a).
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x2
Show that the modulus function f : R → R given by f(x) = |x| is neither one-one nor onto, where |x| is x if x is positive or 0 and |x| is − x if x is negative.
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 3 − 4x
Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is a bijective function.
Let f : R → R be defined as f(x) = 3x. Choose the correct answer.
Let A = {−1, 0, 1} and f = {(x, x2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.
Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2
If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.
Let A = {a, b, c}, B = {u v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as :
f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.
Show that f and g both are bijections and find fog and gof.
Let R+ be the set of all non-negative real numbers. If f : R+ → R+ and g : R+ → R+ are defined as `f(x)=x^2` and `g(x)=+sqrtx` , find fog and gof. Are they equal functions ?
Find fog and gof if : f (x) = |x|, g (x) = sin x .
if f (x) = `sqrt (x +3) and g (x) = x ^2 + 1` be two real functions, then find fog and gof.
State with reason whether the following functions have inverse :
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
Let f be a function from R to R, such that f(x) = cos (x + 2). Is f invertible? Justify your answer.
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.
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 : R → R is defined by f(x) = x2, find f−1 (−25).
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
If the function\[f : R \to \text{A given by} f\left( x \right) = \frac{x^2}{x^2 + 1}\] is a surjection, then A =
The function
\[f : R \to R\] defined by\[f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)\]
(a) one-one but not onto
(b) onto but not one-one
(c) both one and onto
(d) neither one-one nor onto
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
Let
\[f : R \to R\] be a function defined by
Let
\[A = \left\{ x \in R : x \geq 1 \right\}\] The inverse of the function,
\[f : A \to A\] given by
\[f\left( x \right) = 2^{x \left( x - 1 \right)} , is\]
Let
\[A = \left\{ x \in R : x \leq 1 \right\} and f : A \to A\] be defined as
\[f\left( x \right) = x \left( 2 - x \right)\] Then,
\[f^{- 1} \left( x \right)\] is
Mark the correct alternative in the following question:
Let f : R → R be given by f(x) = tanx. Then, f-1(1) is
Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write 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
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}
Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
g = {(1, 4), (2, 4), (3, 4)}
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
If f(x) = (4 – (x – 7)3}, then f–1(x) = ______.
Range of `"f"("x") = sqrt((1 - "cos x") sqrt ((1 - "cos x")sqrt ((1 - "cos x")....infty))`
Let f: R → R defined by f(x) = x4. Choose the correct answer
Consider a function f: `[0, pi/2] ->` R, given by f(x) = sinx and `g[0, pi/2] ->` R given by g(x) = cosx then f and g are
The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.
ASSERTION (A): The relation f : {1, 2, 3, 4} `rightarrow` {x, y, z, p} defined by f = {(1, x), (2, y), (3, z)} is a bijective function.
REASON (R): The function f : {1, 2, 3} `rightarrow` {x, y, z, p} such that f = {(1, x), (2, y), (3, z)} is one-one.
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
