Advertisements
Advertisements
प्रश्न
Let A = {x &epsis; R | −1 ≤ x ≤ 1} and let f : A → A, g : A → A be two functions defined by f(x) = x2 and g(x) = sin (π x/2). Show that g−1 exists but f−1 does not exist. Also, find g−1.
Advertisements
उत्तर
f is not one-one because
f (−1) = (−1)2 = 1
and f (1) = 12 = 1
⇒ -1 and 1 have the same image under f.
⇒ f is not a bijection.
So, f -1 does not exist.
Injectivity of g:
Let x and y be any two elements in the domain (A), such that
g (x) = g (y)
⇒ `sin ((πx)/2) = sin ((πy)/2) `
⇒ `((πx)/2) = ((πy)/2)`
⇒ x = y
So, g is one-one.
Surjectivity of g :
Range of g = ` [ sin ((π(-1))/2) , sin ((π(1))/2) ]`
` = [ sin ((-π)/2) , sin (π/2) ]` = [−1, 1] = A(co-domain of g)
⇒ g is onto.
⇒ g is a bijection.
So, g-1 exists.
Also,
let g−1 (x) = y ...(1)
⇒ g (y) = x
⇒ `sin ((xy)/2) = x`
⇒ `y = 2/π sin^-1 x `
⇒ `g^-1 (x) = 2/π sin^-1 x` [from (1)]
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
Show that the function f : R → {x ∈ R : −1 < x < 1} defined by f(x) = `x/(1 + |x|)`, x ∈ R is one-one and onto function.
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|)
Given examples of two functions f: N → N and g: N → N such that gof is onto but f is not onto.
(Hint: Consider f(x) = x + 1 and `g(x) = {(x-1, ifx >1),(1, if x = 1):}`
Let S = {a, b, c} and T = {1, 2, 3}. Find F−1 of the following functions F from S to T, if it exists.
F = {(a, 3), (b, 2), (c, 1)}
Classify the following function as injection, surjection or bijection :
f : Q → Q, defined by f(x) = x3 + 1
Let A = {1, 2, 3}. Write all one-one from A to itself.
If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.
Show that if f1 and f2 are one-one maps from R to R, then the product f1 × f2 : R → R defined by (f1 × f2) (x) = f1 (x) f2 (x) need not be one - one.
Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.
Find fog and gof if : f (x) = x2 g(x) = cos x .
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 A = {1, 2, 3, 4} and B = {a, b, c, d}, define any four bijections from A to B. Also give their inverse functions.
Which of the following graphs represents a one-one function?
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).
Write the domain of the real function
`f (x) = sqrtx - [x] .`
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
If f(x) = 4 −( x - 7)3 then write f-1 (x).
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
Let
f : R → R be given by
\[f\left( x \right) = \left[ x^2 \right] + \left[ x + 1 \right] - 3\]
where [x] denotes the greatest integer less than or equal to x. Then, f(x) is
(d) one-one and onto
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
Let
\[f : R - \left\{ n \right\} \to R\]
A function f from the set of natural numbers to the set of integers defined by
\[f\left( n \right)\begin{cases}\frac{n - 1}{2}, & \text{when n is odd} \\ - \frac{n}{2}, & \text{when n is even}\end{cases}\]
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
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
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 ______.
For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is injective. Then both f and g are injective functions.
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 A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.
Let f : R `->` R be a function defined by f(x) = x3 + 4, then f is ______.
Range of `"f"("x") = sqrt((1 - "cos x") sqrt ((1 - "cos x")sqrt ((1 - "cos x")....infty))`
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
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 ______.
The domain of function is f(x) = `sqrt(-log_0.3(x - 1))/sqrt(x^2 + 2x + 8)` is ______.
A function f : [– 4, 4] `rightarrow` [0, 4] is given by f(x) = `sqrt(16 - x^2)`. Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a) = `sqrt(7)`.
The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.
ASSERTION (A): The relation f : {1, 2, 3, 4} `rightarrow` {x, y, z, p} defined by f = {(1, x), (2, y), (3, z)} is a bijective function.
REASON (R): The function f : {1, 2, 3} `rightarrow` {x, y, z, p} such that f = {(1, x), (2, y), (3, z)} is one-one.
