Advertisements
Advertisements
Question
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.
Solution
Injectivity of f:
Let x and y be two elements of domain (Q), such that
f(x) = f(y)
⇒">⇒ 2x= 2y
⇒">⇒ x = y
So, f is one-one.
Surjectivity of f:
Let y be in the co-domain (Q), such that f(x) = y.
⇒ 2x = y
⇒ `x = y/2 in Q` (domain)
⇒ is onto.
So, f is a bijection and, hence, it is invertible.
Finding f -1:
Let f−1 (x) =y ...(1)
⇒ x = f (y)
⇒ x = 2y
⇒ `y = x/2`
So, ` f^1 (x) = x/2` (from (1))
njectivity of g:
Let x and y be two elements of domain (Q), such that
g (x) = g (y)
⇒">⇒ x + 2 = y + 2
⇒">⇒ x = y
So, g is one-one.
Surjectivity of g:
Let y be in the co domain (Q), such that g(x) = y.
⇒ x +2 =y
⇒ x= 2 -y ∈ Q (domain)
⇒ g is onto.
So, g is a bijection and, hence, it is invertible.
Finding g -1:
Let g−1(x) = y ...(2)
⇒ x = g (y)
⇒ x = y+2
⇒ y = x − 2
So, g−1 (x) = x − 2 (From (2)
Verification of (gof)−1 = f−1 og −1:
f(x) = 2x ; g (x) = x + 2
and `f^-1 (x) = x/2 ; g^-1 (x)= x-2`
`Now, (f^-1 o g^-1) (x) = f^-1 (g^-1)(x)) `
⇒ `(f^-1 o g ^-1)(x) = f^-1 (x-2) `
⇒ `(f ^-1 o g^-1) (x) = (x-2)/2 .......... (3)`
(gof) (x) = g (f(x))
= g (2x)
= 2x + 2
Let (gof)-1 (x) = y ............ (4)
x = (gof) (y)
⇒ x = 2y +2
⇒ 2y = x - 2
⇒ `y= (x-2)/2`
⇒` (gof)^-1 (x) = (x-2)/2` [form (4) ....... (5) ]
from (3) and (5)
⇒ `(gof)^-1 = f^-1 o g^-1`
APPEARS IN
RELATED QUESTIONS
Let f: N → N be defined by f(n) = `{((n+1)/2, ",if n is odd"),(n/2,",n is even"):}` for all n ∈ N.
State whether the function f is bijective. Justify your answer.
Show that the function f: ℝ → ℝ defined by f(x) = `x/(x^2 + 1), ∀x in R`is neither one-one nor onto. Also, if g: ℝ → ℝ is defined as g(x) = 2x - 1. Find fog(x)
Give an example of a function which is neither one-one nor onto ?
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 : R → R, defined by f(x) = sin2x + cos2x
If f : A → B is an injection, such that range of f = {a}, determine the number of elements in A.
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : `f (x) = x/2`
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.
` if f : (-π/2 , π/2)` → R and g : [−1, 1]→ R be defined as f(x) = tan x and g(x) = `sqrt(1 - x^2)` respectively, describe fog and gof.
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.
Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.
If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
Write whether f : R → R, given by `f(x) = x + sqrtx^2` is one-one, many-one, onto or into.
Write the domain of the real function f defined by f(x) = `sqrt (25 -x^2)` [NCERT EXEMPLAR]
Let f, g : R → R be defined by f(x) = 2x + l and g(x) = x2−2 for all x
∈ R, respectively. Then, find gof. [NCERT EXEMPLAR]
\[f : R \to R \text{given by} f\left( x \right) = x + \sqrt{x^2} \text{ is }\]
The function
f : A → B defined by
f (x) = - x2 + 6x - 8 is a bijection if
Let f be an injective map with domain {x, y, z} and range {1, 2, 3}, such that exactly one of the following statements is correct and the remaining are false.
\[f\left( x \right) = 1, f\left( y \right) \neq 1, f\left( z \right) \neq 2 .\]
The value of
\[f^{- 1} \left( 1 \right)\] is
If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\]
If \[F : [1, \infty ) \to [2, \infty )\] is given by
\[f\left( x \right) = x + \frac{1}{x}, then f^{- 1} \left( x \right)\]
Let \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] Then, for what value of α is \[f \left( f\left( x \right) \right) = x?\]
If f(x) = `(x+3)/(4x−5) , "g"(x) = (3+5x)/(4x−1)` then verify that `("fog") (x)` = x.
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 the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto
Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
Let g(x) = x2 – 4x – 5, then ____________.
If N be the set of all-natural numbers, consider f: N → N such that f(x) = 2x, ∀ x ∈ N, then f is ____________.
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 ____________.
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:
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: {1,2,3,....} → {1,4,9,....} be defined by f(x) = x2 is ____________.
`x^(log_5x) > 5` implies ______.
Let x is a real number such that are functions involved are well defined then the value of `lim_(t→0)[max{(sin^-1 x/3 + cos^-1 x/3)^2, min(x^2 + 4x + 7)}]((sin^-1t)/t)` where [.] is greatest integer function and all other brackets are usual brackets.
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 = R – {2} and B = R – {1}. If f: A `→` B is a function defined by f(x) = `(x - 1)/(x - 2)` then show that f is a one-one and an onto function.
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
The given function f : R → R is not ‘onto’ function. Give reason.
