Advertisements
Advertisements
Question
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
Options
bijection
injection but not a surjection
surjection but not an injection
neither an injection nor a surjection
Advertisements
Solution
\[f\left( x \right) = \sin^{- 1} \left( 3x - 4 x^3 \right)\]
\[ \Rightarrow f\left( x \right) = 3 \sin^{- 1} x\]
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
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x2
Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.
Given examples of two functions f: N → N and g: N → N such that gof is onto but f is not onto.
(Hint: Consider f(x) = x + 1 and `g(x) = {(x-1, ifx >1),(1, if x = 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}.
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, defined by f(x) = x − 5
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sinx
If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x and g(x) = |x| .
Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.
Find fog and gof if : f (x) = x+1, g (x) = sin x .
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
if f (x) = `sqrt (x +3) and g (x) = x ^2 + 1` be two real functions, then find fog and gof.
Find f −1 if it exists : f : A → B, where A = {1, 3, 5, 7, 9}; B = {0, 1, 9, 25, 49, 81} and f(x) = x2
If f : Q → Q, g : Q → Q are two functions defined by f(x) = 2 x and g(x) = x + 2, show that f and g are bijective maps. Verify that (gof)−1 = f−1 og −1.
Let f : R `{- 4/3} `- 43 →">→ R be a function defined as f(x) = `(4x)/(3x +4)` . Show that f : R - `{-4/3}`→ Rang (f) is one-one and onto. Hence, find f -1.
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 : R → R is given by f(x) = x3, write f−1 (1).
If f : R → R defined by f(x) = 3x − 4 is invertible, then write f−1 (x).
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).
If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog. [NCERT EXEMPLAR]
The function \[f : R \to R\] defined by
\[f\left( x \right) = 6^x + 6^{|x|}\] 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
Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is ______.
Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______
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 C be the set of complex numbers. Prove that the mapping f: C → R given by f(z) = |z|, ∀ z ∈ C, is neither one-one nor onto.
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 = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = `(x - 2)/(x - 3)` ∀ x ∈ A . Then show that f is bijective.
Let f: R → R be given by f(x) = tan x. Then f–1(1) is ______.
If f(x) = (4 – (x – 7)3}, then f–1(x) = ______.
Let A = {0, 1} and N be the set of natural numbers. Then the mapping f: N → A defined by f(2n – 1) = 0, f(2n) = 1, ∀ n ∈ N, is onto.
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 ____________.
The function f: R → R defined as f(x) = x3 is:
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?
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.
- Ravi wants to find the number of injective functions from B to G. How many numbers of injective functions are possible?
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.
- The function f: R → R defined by f(x) = x − 4 is ____________.
Let f: R → R defined by f(x) = x4. Choose the correct answer
Function f: R → R, defined by f(x) = `x/(x^2 + 1)` ∀ x ∈ R is not
For \(f(x)=x^2+x\), if \(y=1\), which two domain elements have the same image?
