Advertisements
Advertisements
प्रश्न
Let f be a real function given by f (x)=`sqrt (x-2)`
Find each of the following:
(i) fof
(ii) fofof
(iii) (fofof) (38)
(iv) f2
Also, show that fof ≠ `f^2` .
Advertisements
उत्तर
f (x) = `sqrt(x-2)`
For domain,
x − 2 ≥ 0
⇒ x ≥ 2
Domain of f = [ 2,∞ )
Since f is a square-root function, range of f =( 0,∞)
So, f : [2,∞) → ( 0,∞ )
(i) fof
Range of f is not a subset of the domain of f.
⇒Domain(fof)= { x : x ∈ domain of fand f (x) ∈ domain of f}
⇒ Domain (fof) = `{x :x in [2, ∞ ) and sqrt (x-2) in [ 2 ∞ )}`
⇒ Domain (fof) = `{x :x in [2, ∞ ) and sqrt (x-2)≥ 2 }`
⇒ Domain(fof) = { x : x ∈ [2,∞) and x−2 ≥4 }
⇒ Domain(fof) = { x : x ∈ [2,∞) and x ≥ 6}⇒ Domain(fof) = { x : x ≥ 6}
⇒ Domain(fof) = [ 6, ∞ )
fof : [6, ∞) → R
(fof) (x) = f (f (x))
= ` f (sqrt(x -2))`
= `sqrt (sqrt(x - 2) - 2)`
(ii) fofof= (fof) of
We have, f : [ 2,∞ ) → ( 0,∞ ) and fof : [ 6, ∞ ) → R
⇒ Range of f is not a subset of the domain of fof.
Then, domain((fof)of)={ x : x ∈domain of fand f (x) ∈ domain of fof }
⇒ Domain((fof)of) = `{ x : x in [ 2,∞) and sqrt (x-2) in [ 6 ,∞)}`
⇒ Domain ((fof)of) = ` x:x in [ 2 ∞ ) and sqrt(x-2) ≥ 6 }`
⇒ Domain ((fof)of) = { x : x ∈ [2,∞) and x − 2 ≥ 36}
⇒ Domain ((fof)of) = { x : x ∈ [2,∞) and x ≥ 38 }
⇒ Domain ((fof)of) = { x : x ≥ 38}
⇒ Domain ((fof)of) = [ 38, ∞ )
fof : [38,∞)→ R
So, ((fof)of) (x) = (fof) (f (x))
= (fof) `(sqrt(x-2))`
= `sqrt (sqrt (sqrt(x-2) -2 )-2)`
(iii) We have, (fofof) (x) = `sqrt (sqrt (sqrt(x-2) -2 )-2)`
So, (fofof) (38) = `sqrt (sqrt (sqrt(38-2) -2 )-2)`
=`sqrt (sqrt (sqrt(36) -2 )-2)`
=`sqrt (sqrt(6-2) -2 )`
= `sqrt (2 -2)`
= 0
(iv) We have, fof = `sqrt (sqrt(x-2) -2 )`
` f^2 (x) = f (x) xx f (x) = sqrt(x - 2) xx sqrt(x - 2) = x -2`
So, fof ≠ `f^2`
APPEARS IN
संबंधित प्रश्न
Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Find the number of all onto functions from the set {1, 2, 3, ..., n} to itself.
Give an example of a function which is neither one-one nor onto ?
Which of the following functions from A to B are one-one and onto?
f1 = {(1, 3), (2, 5), (3, 7)} ; A = {1, 2, 3}, B = {3, 5, 7}
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
If A = {1, 2, 3}, show that a one-one function f : A → A must be onto.
Suppose f1 and f2 are non-zero one-one functions from R to R. Is `f_1 / f^2` necessarily one - one? Justify your answer. Here,`f_1/f_2 : R → R is given by (f_1/f_2) (x) = (f_1(x))/(f_2 (x)) for all x in R .`
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
Find fog and gof if : f(x)= x + 1, g (x) = 2x + 3 .
Let f, g, h be real functions given by f(x) = sin x, g (x) = 2x and h (x) = cos x. Prove that fog = go (fh).
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
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.
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.
If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog. [NCERT EXEMPLAR]
Let\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = \text{B and C} = \left\{ x \in R : x \geq 0 \right\} and\]\[S = \left\{ \left( x, y \right) \in A \times B : x^2 + y^2 = 1 \right\} \text{and } S_0 = \left\{ \left( x, y \right) \in A \times C : x^2 + y^2 = 1 \right\}\]
Then,
If the function\[f : R \to \text{A given by} f\left( x \right) = \frac{x^2}{x^2 + 1}\] is a surjection, then A =
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
The function \[f : R \to R\] defined by
\[f\left( x \right) = 6^x + 6^{|x|}\] is
The inverse of the function
\[f : R \to \left\{ x \in R : x < 1 \right\}\] given by
\[f\left( x \right) = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] is
Let \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] Then, for what value of α is \[f \left( f\left( x \right) \right) = x?\]
Mark the correct alternative in the following question:
Let f : R→ R be defined as, f(x) = \[\begin{cases}2x, if x > 3 \\ x^2 , if 1 < x \leq 3 \\ 3x, if x \leq 1\end{cases}\]
Then, find f( \[-\]1) + f(2) + f(4)
Write about strlen() function.
Let R be the set of real numbers and f: R → R be the function defined by f(x) = 4x + 5. Show that f is invertible and find f–1.
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:
g(x) = |x|
If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is ______.
Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
The function f : R → R given by f(x) = x3 – 1 is ____________.
If N be the set of all-natural numbers, consider f: N → N such that f(x) = 2x, ∀ x ∈ N, then f is ____________.
Let f: R → R defined by f(x) = 3x. Choose the correct answer
Let f(x) = ax (a > 0) be written as f(x) = f1(x) + f2(x), where f1(x) is an even function and f2(x) is an odd function. Then f1(x + y) + f1(x – y) equals ______.
Let f(x) be a polynomial of degree 3 such that f(k) = `-2/k` for k = 2, 3, 4, 5. Then the value of 52 – 10f(10) is equal to ______.
Find the domain of sin–1 (x2 – 4).
A function is called one-one if different elements of the domain have:
For \(f(x)=x^2+x\), if \(y=1\), which two domain elements have the same image?
