Advertisements
Advertisements
प्रश्न
Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but gis not injective.
Advertisements
उत्तर
Let f : N → Z be given by f (x) = x, which is injective.
(If we take f(x) = f(y), then it gives x = y)
Let g : Z → Z be given by g (x) = |x|, which is not injective.
If we take f(x) = f(y), we get:
|x| = |y|
⇒ x = ± y
Now, gof : N → Z.
(gof) (x)=g (f (x)) = g (x) = |x|
Let us take two elements x and y in the domain of gof , such that
(gof) (x) = (gof) (y)
⇒ |x| = |y|
⇒ x = y (We don't get ± here because x, y ∈N)
So, gof is injective.
APPEARS IN
संबंधित प्रश्न
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 f : N → N be defined by f(n) = `{((n+1)/2", if n is odd"),(n/2", if n is even"):}` for all n ∈ N.
State whether the function f is bijective. Justify your answer.
Classify the following function as injection, surjection or bijection :
f : Q → Q, defined by f(x) = x3 + 1
Show that the function f : R − {3} → R − {2} given by f(x) = `(x-2)/(x-3)` is a bijection.
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:
(i) an injective map from A to B
(ii) a mapping from A to B which is not injective
(iii) a mapping from A to B.
Find fog and gof if : f (x) = ex g(x) = loge x .
Find fog and gof if : f (x) = x2 g(x) = cos x .
Find fog and gof if : f (x) = x+1, 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.
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
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 : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
Let f : R − {−1} → R − {1} be given by\[f\left( x \right) = \frac{x}{x + 1} . \text{Write } f^{- 1} \left( x \right)\]
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]
If the mapping f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3}, given by f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, then write fog. [NCERT EXEMPLAR]
Let\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = \text{B and C} = \left\{ x \in R : x \geq 0 \right\} and\]\[S = \left\{ \left( x, y \right) \in A \times B : x^2 + y^2 = 1 \right\} \text{and } S_0 = \left\{ \left( x, y \right) \in A \times C : x^2 + y^2 = 1 \right\}\]
Then,
\[f : R \to R \text{given by} f\left( x \right) = x + \sqrt{x^2} \text{ is }\]
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
The function
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.
The smallest integer function f(x) = [x] is ____________.
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
The domain of the function `"f"("x") = 1/(sqrt ({"sin x"} + {"sin" ( pi + "x")}))` where {.} denotes fractional part, is
Range of `"f"("x") = sqrt((1 - "cos x") sqrt ((1 - "cos x")sqrt ((1 - "cos x")....infty))`
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever
Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation ‘R’ is defined on I as follows:
R = {(V1, V2) ∶ V1, V2 ∈ I and both use their voting right in the general election - 2019}
- Three friends F1, F2, and F3 exercised their voting right in general election-2019, then which of the following is true?
Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.
Answer the following questions using the above information.
- Let f: R → R be defined by f(x) = x2 is:
If `f : R -> R^+ U {0}` be defined by `f(x) = x^2, x ∈ R`. The mapping is
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
Let [x] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)` is (–∞, a) ∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is ______.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
Consider a set containing function A= {cos–1cosx, sin(sin–1x), sinx((sinx)2 – 1), etan{x}, `e^(|cosx| + |sinx|)`, sin(tan(cosx)), sin(tanx)}. B, C, D, are subsets of A, such that B contains periodic functions, C contains even functions, D contains odd functions then the value of n(B ∩ C) + n(B ∩ D) is ______ where {.} denotes the fractional part of functions)
If f: [0, 1]→[0, 1] is defined by f(x) = `(x + 1)/4` and `d/(dx) underbrace(((fofof......of)(x)))_("n" "times")""|_(x = 1/2) = 1/"m"^"n"`, m ∈ N, then the value of 'm' is ______.
Let f: R→R be a polynomial function satisfying f(x + y) = f(x) + f(y) + 3xy(x + y) –1 ∀ x, y ∈ R and f'(0) = 1, then `lim_(x→∞)(f(2x))/(f(x)` is equal to ______.
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)
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
Let f: R→Rbe defined as f (x) = `(x^2 + 1)/2`, then ______.
The function \(f:\mathbb{N}\to\mathbb{N}\) defined by \(f(x)=5x\) is:
