Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Let f : R → R be defined as f(x) = x4. Choose the correct answer.
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, 2), (b, 1), (c, 1)}
Give an example of a function which is one-one but not onto ?
Give an example of a function which is neither one-one nor onto ?
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 1 + x2
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + 3 and g(x) = x2 + 5 .
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.
Find fog and gof if : f(x) = sin−1 x, g(x) = x2
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.
If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x|- x, ∀x∈R" .Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2).
Find f −1 if it exists : f : A → B, where A = {0, −1, −3, 2}; B = {−9, −3, 0, 6} and f(x) = 3 x.
If f : R → (−1, 1) defined by `f (x) = (10^x- 10^-x)/(10^x + 10 ^-x)` is invertible, find f−1.
Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.
If A = {a, b, c} and B = {−2, −1, 0, 1, 2}, write the total number of one-one functions from A to B.
Let f : R − {−1} → R − {1} be given by\[f\left( x \right) = \frac{x}{x + 1} . \text{Write } f^{- 1} \left( x \right)\]
Write the domain of the real function
`f (x) = sqrtx - [x] .`
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
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,
Let
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = B\] Then, the mapping\[f : A \to \text{B given by} f\left( x \right) = x\left| x \right|\] is
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
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 \[f : R \to \left( - 1, 1 \right)\] is defined by
\[f\left( x \right) = \frac{- x|x|}{1 + x^2}, \text{ then } f^{- 1} \left( x \right)\] equals
Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f−1.
Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
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 C be the set of complex numbers. Prove that the mapping f: C → R given by f(z) = |z|, ∀ z ∈ C, is neither one-one nor onto.
Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
Let f : R `->` R be a function defined by f(x) = x3 + 4, then f is ______.
Let f : R → R, g : R → R be two functions such that f(x) = 2x – 3, g(x) = x3 + 5. The function (fog)-1 (x) is equal to ____________.
Range of `"f"("x") = sqrt((1 - "cos x") sqrt ((1 - "cos x")sqrt ((1 - "cos x")....infty))`
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Let R: B → G be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then R is ____________.
If f; R → R f(x) = 10x + 3 then f–1(x) is:
Prove that the function f is surjective, where f: N → N such that `f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if "n is even"):}` Is the function injective? Justify your answer.
Let f: R→R be defined as f(x) = 2x – 1 and g: R – {1}→R be defined as g(x) = `(x - 1/2)/(x - 1)`. Then the composition function f (g(x)) is ______.
Let f(x) be a polynomial function of degree 6 such that `d/dx (f(x))` = (x – 1)3 (x – 3)2, then
Assertion (A): f(x) has a minimum at x = 1.
Reason (R): When `d/dx (f(x)) < 0, ∀ x ∈ (a - h, a)` and `d/dx (f(x)) > 0, ∀ x ∈ (a, a + h)`; where 'h' is an infinitesimally small positive quantity, then f(x) has a minimum at x = a, provided f(x) is continuous at x = a.
