Advertisements
Advertisements
प्रश्न
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
Advertisements
उत्तर
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
It can be written as,
f (x) = `{ (1 +x , 0 ≤ x ≤ 1) , (1 +x, 1< x ≤ 2) ,( 3 - x, 2 < x ≤ 3):}`
When, 0 ≤ x ≤ 1
Then , `f (x) = 1 +x `
Now when , 0 ≤ x ≤ 1 then ,1 ≤ x + 1 ≤ 2
Then , `f (f(x))` = 1 + (1 + x ) = 2 + x [ ∵ 1 ≤ f (x) < 2]
When ,1 < x ≤ 2
Then , f (x) = 1 + x
Now when , 1 < x ≤ 2 then,2 < x +1 ≤ 3
Then , f (f(x)) = 3 − ( 1+ x ) = 2 − x [ ∵ 2 ≤ f(x) <3 ]
When , 2 < x ≤ 3
Then , f (x) = 3 - x
Now when ,2< x ≤ 3 then ,0 ≤ 3 − x < 1
Then , f (f(x)) = 1 + ( 3 − x ) = 4 − x [ ∵ 0 ≤ f (x) < 1 ]
f(f(x)) = ` {(2 + x , 0 ≤ x ≤ 1) , (2 -x, 1 < x ≤ 2),( 4- x , 2 < x ≤ 3):}`
APPEARS IN
संबंधित प्रश्न
Show that the signum function f : R → R, given by
`f(x) = {(1", if" x > 0), (0", if" x = 0), (-1", if" x < 0):}`
is neither one-one nor onto.
Let f: R → R be the Signum Function defined as
f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`
and g: R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0, 1]?
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x2
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x − 5
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.
Verify associativity for the following three mappings : f : N → Z0 (the set of non-zero integers), g : Z0 → Q and h : Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = ex.
Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.
if `f (x) = sqrt(1-x)` and g(x) = `log_e` x are two real functions, then describe functions fog and gof.
If f : R → (0, 2) defined by `f (x) =(e^x - e^(x))/(e^x +e^(-x))+1`is invertible , find f-1.
Let f be a function from C (set of all complex numbers) to itself given by f(x) = x3. Write f−1 (−1).
If f : C → C is defined by f(x) = (x − 2)3, write f−1 (−1).
Let `f : R - {- 3/5}` → R be a function defined as `f (x) = (2x)/(5x +3).`
f-1 : Range of f → `R -{-3/5}`.
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 is defined by f(x) = 3x + 2, find f (f (x)).
If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = \[\alpha x + \beta\] then find the values of \[\alpha\] and \[ \beta\] . [NCERT EXEMPLAR]
The function
f : A → B defined by
f (x) = - x2 + 6x - 8 is a bijection if
Let
\[f : R \to R\] be a function defined by
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
Let
\[A = \left\{ x \in R : x \geq 1 \right\}\] The inverse of the function,
\[f : A \to A\] given by
\[f\left( x \right) = 2^{x \left( x - 1 \right)} , is\]
If the function
\[f : R \to R\] be such that
\[f\left( x \right) = x - \left[ x \right]\] where [x] denotes the greatest integer less than or equal to x, then \[f^{- 1} \left( x \right)\]
If \[F : [1, \infty ) \to [2, \infty )\] is given by
\[f\left( x \right) = x + \frac{1}{x}, then f^{- 1} \left( x \right)\]
The distinct linear functions that map [−1, 1] onto [0, 2] are
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
Let the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto
Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
h = {(1,4), (2, 5), (3, 5)}
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.
Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R then 'f' 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.
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)`.
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
The given function f : R → R is not ‘onto’ function. Give reason.
A function is onto when every element of the codomain has:
A function is called bijective if it is both:
Which real-life situation resembles a one-one idea when no roll number is repeated?
