Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let A = R – {2}, B = R – {1}
f: A `→` B s.t. f(x) = `(x - 1)/(x - 2)`
For x1, x2 ∈ A
f(x1) = f(x2)
`\implies (x_1 - 1)/(x_1 - 2) = (x_2 - 1)/(x_2 - 2)`
`\implies (x_1 - 1)/(x_1 - 2) - 1 = (x_2 - 1)/(x_2 - 2) - 1`
`\implies 1/(x_1 - 2) = 1/(x_2 - 2)`
`\implies` x1 – 2 = x2 – 2
`\implies` x1 = x2
∴ f(x) is one-one function.
Also if f(x) = y, where y ∈ B.
`\implies (x - 1)/(x - 2)` = y
`\implies` x – 1 = xy – 2y
`\implies` 2y – 1 = xy – x
`\implies` 2y – 1 = x(y – 1)
x = `(2y - 1)/(y - 1) ∈ A`
Clearly every element y ∈ B is associated to x = `(2y - 1)/(y - 1)` of set A.
So Range of f = B `\implies` f is into
Hence f is one-one and onto function.
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.
Show that the function f : R → R given by f(x) = x3 is injective.
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sin2x + cos2x
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 : A → B and g : B → C are onto functions, show that gof is a onto function.
State with reason whether the following functions have inverse :
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
If a function\[f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5\] is a bijection, then B =
\[f : Z \to Z\] be given by
` f (x) = {(x/2, ", if x is even" ) ,(0 , ", if x is odd "):}`
Then, f is
The function \[f : R \to R\] defined by
\[f\left( x \right) = 6^x + 6^{|x|}\] is
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 \[f : R \to \left( - 1, 1 \right)\] is defined by
\[f\left( x \right) = \frac{- x|x|}{1 + x^2}, \text{ then } f^{- 1} \left( x \right)\] equals
If \[f : R \to R\] is given by \[f\left( x \right) = x^3 + 3, \text{then} f^{- 1} \left( x \right)\] is equal to
Mark the correct alternative in the following question:
Let f : R → R be given by f(x) = tanx. Then, f-1(1) is
Mark the correct alternative in the following question:
If the set A contains 7 elements and the set B contains 10 elements, then the number one-one functions from A to B is
Write about strlen() function.
`x^(log_5x) > 5` implies ______.
Let a and b are two positive integers such that b ≠ 1. Let g(a, b) = Number of lattice points inside the quadrilateral formed by lines x = 0, y = 0, x = b and y = a. f(a, b) = `[a/b] + [(2a)/b] + ... + [((b - 1)a)/b]`, then the value of `[(g(101, 37))/(f(101, 37))]` is ______.
(Note P(x, y) is lattice point if x, y ∈ I)
(where [.] denotes greatest integer function)
Which condition represents a many-one function?
