Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
A = R – {3}, B = R – {1}
f : A → B is defined as f(x) = `((x - 2)/(x - 3))`
Let x, y ∈ A such that f(x) = f(y)
⇒ `(x - 2)/(x - 3) = (y - 2)/(y - 3)`
⇒ (x – 2) (y – 3) = (y – 2) (x – 3)
⇒ xy – 3x – 2y + 6 = xy – 3y – 2x + 6
⇒ –3x – 2y = –3y – 2x
⇒ 3x – 2x = 3y – 2y
⇒ x = y
∴ f is one-one.
Let y ∈ B = R – {1}. 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(1 – y) = –3y + 2
⇒ `x = (2 - 3y)/(1- y) ∈ 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.
Hence, function f is one-one and onto.
APPEARS IN
संबंधित प्रश्न
Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.
Let f : R → R be defined as f(x) = 3x. Choose the correct 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.
Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but gis not injective.
(Hint: Consider f(x) = x and g(x) =|x|)
Give an example of a function which is one-one but not onto ?
Give an example of a function which is not one-one but onto ?
Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x2
Show that the exponential function f : R → R, given by f(x) = ex, is one-one but not onto. What happens if the co-domain is replaced by`R0^+` (set of all positive real numbers)?
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
Let f : N → N be defined by
`f(n) = { (n+ 1, if n is odd),( n-1 , if n is even):}`
Show that f is a bijection.
[CBSE 2012, NCERT]
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + x2 and g(x) = x3
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x2 + 8 and g(x) = 3x3 + 1 .
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x and g(x) = |x| .
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x2 + 2x − 3 and g(x) = 3x − 4 .
Let f = {(1, −1), (4, −2), (9, −3), (16, 4)} and g = {(−1, −2), (−2, −4), (−3, −6), (4, 8)}. Show that gof is defined while fog is not defined. Also, find gof.
Let A = {a, b, c}, B = {u v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as :
f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.
Show that f and g both are bijections and find fog and gof.
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
If A = {1, 2, 3} and B = {a, b}, write the total number of functions from A to B.
If A = {a, b, c} and B = {−2, −1, 0, 1, 2}, write the total number of one-one functions from A to B.
If f : R → R is defined by f(x) = x2, find f−1 (−25).
Let f : R → R, g : R → R be two functions defined by f(x) = x2 + x + 1 and g(x) = 1 − x2. Write fog (−2).
If f : R → R be defined by f(x) = (3 − x3)1/3, then find fof (x).
Let f : R → R be the function defined by f(x) = 4x − 3 for all x ∈ R Then write f . [NCERT EXEMPLAR]
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
Let
\[A = \left\{ x : - 1 \leq x \leq 1 \right\} \text{and} f : A \to \text{A such that f}\left( x \right) = x|x|\]
A function f from the set of natural numbers to the set of integers defined by
\[f\left( n \right)\begin{cases}\frac{n - 1}{2}, & \text{when n is odd} \\ - \frac{n}{2}, & \text{when n is even}\end{cases}\]
\[f : Z \to Z\] be given by
` f (x) = {(x/2, ", if x is even" ) ,(0 , ", if x is odd "):}`
Then, f is
The distinct linear functions that map [−1, 1] onto [0, 2] are
If f: R → R is defined by f(x) = x2 – 3x + 2, write f(f (x))
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
h(x) = x|x|
Let R be a relation on the set L of lines defined by l1 R l2 if l1 is perpendicular to l2, then relation R is ____________.
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.
- Let R: B → G be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then R is ____________.
Let f: R → R defined by f(x) = 3x. Choose the correct answer
If `f : R -> R^+ U {0}` be defined by `f(x) = x^2, x ∈ R`. The mapping is
Consider a set containing function A= {cos–1cosx, sin(sin–1x), sinx((sinx)2 – 1), etan{x}, `e^(|cosx| + |sinx|)`, sin(tan(cosx)), sin(tanx)}. B, C, D, are subsets of A, such that B contains periodic functions, C contains even functions, D contains odd functions then the value of n(B ∩ C) + n(B ∩ D) is ______ where {.} denotes the fractional part of functions)
Find the domain of sin–1 (x2 – 4).
In the function from \(X\) to \(Y\), if \(6\in Y\) is not the image of any element of \(X\), the function is:
A perfect one-to-one matching between users and unique IDs is similar to:
Which condition represents an into function?
