Advertisements
Advertisements
प्रश्न
Let A = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = `(x - 2)/(x - 3)` ∀ x ∈ A . Then show that f is bijective.
Advertisements
उत्तर
Given that,
A = R – {3}, B = R – {1}
f: A → B be defined by f(x) = `(x - 2)/x` – 3 ∀ x ∈ A
∴ f(x) = `(x - 3 + 1)/(x - 3)` = 1 + `1/(x - 3)`
Let f(x1) = f (x2)
⇒ `1 + 1/(x_1 - 3) = 1 + 1/(x_2 - 3)`
⇒ `1/(x_1 - 3) = 1/(x_2 - 3)`
⇒ x1 = x2
So, (x) is an injection function
Now let y = `(x - 2)/(x - 3)`
⇒ x – 2 = xy – 3y
⇒ x(1 – y) = 2 – 3y
⇒ x = `(2 - 3y)/(1 - y)`
⇒ x = `(3y - 2)/(y - 1)`
⇒ y ∈ R – {1} = B
Since y ∈ R – {1} = B for every y ∈ B, there exists an x ∈ A such that f(x) = y.
Therefore, f(x) is surjective (onto).
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x2
Give an example of a function which is one-one but not onto ?
Which of the following functions from A to B are one-one and onto ?
f3 = {(a, x), (b, x), (c, z), (d, z)} ; A = {a, b, c, d,}, B = {x, y, z}.
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sinx
Show that the function f : R − {3} → R − {2} given by f(x) = `(x-2)/(x-3)` is a bijection.
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:
(i) an injective map from A to B
(ii) a mapping from A to B which is not injective
(iii) a mapping from A to B.
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
If f : R → R be defined by f(x) = x3 −3, then prove that f−1 exists and find a formula for f−1. Hence, find f−1(24) and f−1 (5).
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.
Consider the function f : R+ → [-9 , ∞ ]given by f(x) = 5x2 + 6x - 9. Prove that f is invertible with f -1 (y) = `(sqrt(54 + 5y) -3)/5` [CBSE 2015]
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.
If f : C → C is defined by f(x) = x4, write f−1 (1).
Which of the following functions form Z to itself are bijections?
The distinct linear functions that map [−1, 1] onto [0, 2] are
Mark the correct alternative in the following question:
Let f : R \[-\] \[\left\{ \frac{3}{5} \right\}\] \[\to\] R be defined by f(x) = \[\frac{3x + 2}{5x - 3}\] Then,
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.
Which function is used to check whether a character is alphanumeric or not?
Write about strlen() function.
Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is ______.
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}
Let the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
h(x) = x|x|
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
Which of the following functions from Z into Z are bijections?
Let f: `[2, oo)` → R be the function defined by f(x) = x2 – 4x + 5, then the range of f is ______.
Let f: R → R be given by f(x) = tan x. Then f–1(1) is ______.
The function f : A → B defined by f(x) = 4x + 7, x ∈ R is ____________.
The smallest integer function f(x) = [x] 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.
- The function f: Z → Z defined by f(x) = x2 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(1, 3) `rightarrow` R be a function defined by f(x) = `(x[x])/(1 + x^2)`, where [x] denotes the greatest integer ≤ x, Then the range of f is ______.
Find the domain of sin–1 (x2 – 4).
A function is called bijective if it is both:
Let \(X=\{-1,2,5,8,11\}\), \(Y=\{4,6,8\}\), and suppose different elements \(2,5,8\) of \(X\) have the same image \(4\). What type of function is it with respect to the one-one property?
In the function from \(X\) to \(Y\), if \(6\in Y\) is not the image of any element of \(X\), the function is:
