Advertisements
Advertisements
Question
The function
f : A → B defined by
f (x) = - x2 + 6x - 8 is a bijection if
Options
A = (- ∞ , 3] and B = ( - ∞, 1 ]
A = [- 3 , ∞) and B = ( - ∞, 1 ]
A = (- ∞ , 3] and B = [ 1 ,∞)
A = [3 ,∞ ) and B = [ 1 ,∞ )
Advertisements
Solution
\[A = ( - \infty , 3] \text{and }B = ( - \infty , 1]\]
\[f\left( x \right) = - x^2 + 6x - 8 , \text{is a polynomial function}\]
\[\text{And the domain of polynomial function is real number} . \]
\[ \therefore x \in R\]
\[f(x) = - x^2 + 6x - 8\]
\[ = - \left( x^2 - 6x + 8 \right)\]
\[ = - \left( x^2 - 6x + 9 - 1 \right)\]
\[ = - \left( x - 3 \right)^2 + 1\]
\[\text{Maximum value of} - \left( x - 3 \right)^2 \text{woud be } 0\]
\[ \therefore \text{Maximum value of} - \left( x - 3 \right)^2 + 1 \text{woud be} 1\]
\[ \therefore f(x) \in ( - \infty , 1]\]
\[\text{We can see from the given graph that function is symmetrical about x = 3 & the given function is bijective .} \]
\[\text{So, x would be either} ( - \infty , 3 ] or [ 3, \infty )\]
\[\text{The correct option which satisfy A and B both is}: \]
\[A = ( - \infty , 3] \text{ and }B = ( - \infty , 1]\]
APPEARS IN
RELATED QUESTIONS
Show that the function f in `A=R-{2/3} ` defined as `f(x)=(4x+3)/(6x-4)` is one-one and onto hence find f-1
Show that the signum function f : R → R, given by
`f(x) = {(1", if" x > 0), (0", if" x = 0), (-1", if" x < 0):}`
is neither one-one nor onto.
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. Show that f is one-one.
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)}
Give an example of a function which is not one-one but onto ?
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 : R → R, defined by f(x) = x3 − x
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sin2x + cos2x
If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.
If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.
Let f : R → R and g : R → R be defined by f(x) = x + 1 and g (x) = x − 1. Show that fog = gof = IR.
Consider f : N → N, g : N → N and h : N → R defined as f(x) = 2x, g(y) = 3y + 4 and h(z) = sin z for all x, y, z ∈ N. Show that ho (gof) = (hog) of.
Find fog and gof if : f (x) = |x|, g (x) = sin x .
Find fog and gof if : f (x) = x+1, g(x) = `e^x`
.
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
Which one of the following graphs represents a function?
If f : R → R is defined by f(x) = x2, write f−1 (25)
Let f : R → R be defined as `f (x) = (2x - 3)/4.` write fo f-1 (1) .
Which of the following functions from
to itself are bijections?
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
A function f from the set of natural numbers to the set of integers defined by
\[f\left( n \right)\begin{cases}\frac{n - 1}{2}, & \text{when n is odd} \\ - \frac{n}{2}, & \text{when n is even}\end{cases}\]
If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\]
Let
\[A = \left\{ x \in R : x \geq 1 \right\}\] The inverse of the function,
\[f : A \to A\] given by
\[f\left( x \right) = 2^{x \left( x - 1 \right)} , is\]
If \[F : [1, \infty ) \to [2, \infty )\] is given by
\[f\left( x \right) = x + \frac{1}{x}, then f^{- 1} \left( x \right)\]
If \[f : R \to \left( - 1, 1 \right)\] is defined by
\[f\left( x \right) = \frac{- x|x|}{1 + x^2}, \text{ then } f^{- 1} \left( x \right)\] equals
A function f: R→ R defined by f(x) = `(3x) /5 + 2`, x ∈ R. Show that f is one-one and onto. Hence find f−1.
Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.
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
k = {(1,4), (2, 5)}
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
k(x) = x2
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}
- Mr. ’X’ and his wife ‘W’ both exercised their voting right in the general election-2019, Which of the following is true?
Prove that the function f is surjective, where f: N → N such that `f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if "n is even"):}` Is the function injective? Justify your answer.
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.
The graph of the function y = f(x) is symmetrical about the line x = 2, then ______.
If A = {x ∈ R: |x – 2| > 1}, B = `{x ∈ R : sqrt(x^2 - 3) > 1}`, C = {x ∈ R : |x – 4| ≥ 2} and Z is the set of all integers, then the number of subsets of the set (A ∩ B ∩ C) C ∩ Z is ______.
The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.
A function is a rule that assigns each element of one set to exactly how many elements of another set?
For \(f:\mathbb{R}\to\mathbb{R}\) defined by \(f(x)=1+x^2\), which statement is correct?
A perfect one-to-one matching between users and unique IDs is similar to:
