Advertisements
Advertisements
प्रश्न
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
विकल्प
bijection
injection but not a surjection
surjection but not an injection
neither an injection nor a surjection
Advertisements
उत्तर
\[f\left( x \right) = \sin^{- 1} \left( 3x - 4 x^3 \right)\]
\[ \Rightarrow f\left( x \right) = 3 \sin^{- 1} x\]
Injectivity:
Let x and y be two elements in the domain
\[\left[ \frac{- 1}{2}, \frac{1}{2} \right]\] , such that
\[f\left( x \right) = f\left( y \right)\]
\[ \Rightarrow 3 \sin^{- 1} x = 3 \sin^{- 1} y\]
\[ \Rightarrow \sin^{- 1} x = \sin^{- 1} y\]
\[ \Rightarrow x = y\]
So, f is one-one.
Surjectivity:
Let y be any element in the co-domain, such that
\[f\left( x \right) = y\]
\[ \Rightarrow 3 \sin^{- 1} \left( x \right) = y\]
\[ \Rightarrow \sin^{- 1} \left( x \right) = \frac{y}{3}\]
\[ \Rightarrow x = \sin\frac{y}{3} \in \left[ \frac{- 1}{2}, \frac{1}{2} \right]\]
\[\Rightarrow\] f is onto.
\[\Rightarrow\] f is a bijection.
So, the answer is (a).
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) = 1 + x2
Let f: R → R be the Signum Function defined as
f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`
and g: R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0, 1]?
Which of the following functions from A to B are one-one and onto ?
f3 = {(a, x), (b, x), (c, z), (d, z)} ; A = {a, b, c, d,}, B = {x, y, z}.
Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x2
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x − 5
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : `f (x) = x/2`
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : g(x) = |x|
Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2
Let A = {1, 2, 3}. Write all one-one from A to itself.
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 (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
If f : A → B and g : B → C are onto functions, show that gof is a onto function.
Find fog and gof if : f(x)= x + 1, g (x) = 2x + 3 .
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
If f : R → R is defined by f(x) = x2, write f−1 (25)
Write the domain of the real function
`f (x) = sqrtx - [x] .`
\[f : R \to R\] is defined by
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is\]
\[f : Z \to Z\] be given by
` f (x) = {(x/2, ", if x is even" ) ,(0 , ", if x is odd "):}`
Then, f is
Let \[f\left( x \right) = \frac{1}{1 - x} . \text{Then}, \left\{ f o \left( fof \right) \right\} \left( x \right)\]
Mark the correct alternative in the following question:
Let f : R → R be given by f(x) = tanx. Then, f-1(1) is
Mark the correct alternative in the following question:
Let f : R→ R be defined as, f(x) = \[\begin{cases}2x, if x > 3 \\ x^2 , if 1 < x \leq 3 \\ 3x, if x \leq 1\end{cases}\]
Then, find f( \[-\]1) + f(2) + f(4)
Mark the correct alternative in the following question:
Let f : R \[-\] \[\left\{ \frac{3}{5} \right\}\] \[\to\] R be defined by f(x) = \[\frac{3x + 2}{5x - 3}\] 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.
If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one-one from A onto A. 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 f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1
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)}
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
h(x) = x|x|
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- Let f: R → R be defined by f(x) = x − 4. Then the range of f(x) is ____________.
Let f: R → R defined by f(x) = 3x. Choose the correct answer
A function f: x → y is/are called onto (or surjective) if x under f.
`x^(log_5x) > 5` implies ______.
Let f(x) = ax (a > 0) be written as f(x) = f1(x) + f2(x), where f1(x) is an even function and f2(x) is an odd function. Then f1(x + y) + f1(x – y) equals ______.
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 ______.
Write the domain and range (principle value branch) of the following functions:
f(x) = tan–1 x.
The given function f : R → R is not ‘onto’ function. Give reason.
