Advertisements
Advertisements
Question
Let A = R - {3} and B = R - {1}. Consider the function f : A → B defined by f(x) = `(x-2)/(x-3).`Show that f is one-one and onto and hence find f-1.
[CBSE 2012, 2014]
Advertisements
Solution
We have,
A = R {3} and B = R - {1}
The function f : A → B defined by f(x) = `(x-2)/(x-3)`
Let x,y ∈ A such that f (x) = f (y). Then,
`(x-2)/(x-3) = (y-2)/(y-3)`
⇒ xy −3x − 2y + 6 = xy − 2x − 3y + 6
⇒ −x = −y
⇒ x = y
∴ f is one − one .
Let y ∈ B. Then, y ≠ 1.
The function f is onto if there exists x ∈ A such that f (x) = y . Now,
f (x) = y
⇒ `(x-2)/(x-3) = y`
⇒ x - 2 = xy - 3y
⇒ x - xy = 2 - 3y
⇒ x (1 - y ) = 2 - 3y
⇒ ` x = (2 - 3y)/ (1 - y ) in A [ y ≠ 1]`
Thus, for any y ∈ B, there exists `(2-3y)/(1-y)` ∈ A such that
`f ((2 - 3y) / (1-y)) = (((2-3y)/(1-y))-2)/(((2-3y)/(1 - y))-3 `= `(2-3y - 2 +2y)/(2-3y -3 +3y) = (-y)/(-1) = y`
∴ f is onto.
So, f is one−one and onto fucntion.
Now,
` As , x = (2 - 3y)/(1- y)`
`so , f^-1 (x) = (2- 3x)/(1 - x) = (3x - 2)/(x-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
Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.
Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by f(x) = `((x - 2)/(x - 3))`. Is f one-one and onto? Justify your answer.
Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1R.
Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x3
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
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 : Q − {3} → Q, defined by `f (x) = (2x +3)/(x-3)`
Show that the function f : R − {3} → R − {2} given by f(x) = `(x-2)/(x-3)` is a bijection.
If A = {1, 2, 3}, show that a one-one function f : A → A must be onto.
Give examples of two one-one functions f1 and f2 from R to R, such that f1 + f2 : R → R. defined by (f1 + f2) (x) = f1 (x) + f2 (x) is not one-one.
Suppose f1 and f2 are non-zero one-one functions from R to R. Is `f_1 / f^2` necessarily one - one? Justify your answer. Here,`f_1/f_2 : R → R is given by (f_1/f_2) (x) = (f_1(x))/(f_2 (x)) for all x in R .`
Let R+ be the set of all non-negative real numbers. If f : R+ → R+ and g : R+ → R+ are defined as `f(x)=x^2` and `g(x)=+sqrtx` , find fog and gof. Are they equal functions ?
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
Find fog and gof if : f (x) = x2 g(x) = cos x .
If f(x) = 2x + 5 and g(x) = x2 + 1 be two real functions, then describe each of the following functions:
(1) fog
(2) gof
(3) fof
(4) f2
Also, show that fof ≠ f2
Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.
Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) = cat. Show that f, g and gof are invertible. Find f−1, g−1 and gof−1and show that (gof)−1 = f −1o g−1
A function f : R → R is defined as f(x) = x3 + 4. Is it a bijection or not? In case it is a bijection, find f−1 (3).
Let f : [−1, ∞) → [−1, ∞) be given by f(x) = (x + 1)2 − 1, x ≥ −1. Show that f is invertible. Also, find the set S = {x : f(x) = f−1 (x)}.
If f : R → R is defined by f(x) = x2, write f−1 (25)
Let f be an invertible real function. Write ( f-1 of ) (1) + ( f-1 of ) (2) +..... +( f-1 of ) (100 )
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,
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 \[g\left( x \right) = x^2 + x - 2\text{ and} \frac{1}{2} gof\left( x \right) = 2 x^2 - 5x + 2\] is equal to
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 N be the set of natural numbers and the function f: N → N be defined by f(n) = 2n + 3 ∀ n ∈ N. Then f 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.
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(x, y): x is a person, y is the mother of x}
Let g(x) = x2 – 4x – 5, then ____________.
Given a function If as f(x) = 5x + 4, x ∈ R. If g : R → R is inverse of function ‘f then
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?
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 ____________.
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R then 'f' is
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.
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.
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 ______.
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
