UrbanPro
true

Find the best tutors and institutes for Class 12 Tuition

Find Best Class 12 Tuition

Please select a Category.

Please select a Locality.

No matching category found.

No matching Locality found.

Outside India?

Learn Exercise 1.3 with Free Lessons & Tips

Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by = {(1, 2), (3, 5), (4, 1)} and = {(1, 3), (2, 3), (5, 1)}. Write down gof.

The functions f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} are defined as

f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}.

Comments

Let fg and h be functions from to R. Show that

To prove:

Comments

Find goand fog, if

(i)

(ii)

(i)

(ii)

Comments

If, show that f f(x) = x, for all. What is the inverse of f?

It is given that.

Hence, the given function f is invertible and the inverse of f is f itself.

Comments

State with reason whether following functions have inverse

(i) f: {1, 2, 3, 4} → {10} with

f = {(1, 10), (2, 10), (3, 10), (4, 10)}

(ii) g: {5, 6, 7, 8} → {1, 2, 3, 4} with

g = {(5, 4), (6, 3), (7, 4), (8, 2)}

(iii) h: {2, 3, 4, 5} → {7, 9, 11, 13} with

h = {(2, 7), (3, 9), (4, 11), (5, 13)}

(i) f: {1, 2, 3, 4} → {10}defined as:

f = {(1, 10), (2, 10), (3, 10), (4, 10)}

From the given definition of f, we can see that f is a many one function as: f(1) = f(2) = f(3) = f(4) = 10

f is not one-one.

Hence, function f does not have an inverse.

(ii) g: {5, 6, 7, 8} → {1, 2, 3, 4} defined as:

g = {(5, 4), (6, 3), (7, 4), (8, 2)}

From the given definition of g, it is seen that g is a many one function as: g(5) = g(7) = 4.

g is not one-one,

Hence, function g does not have an inverse.

(iii) h: {2, 3, 4, 5} → {7, 9, 11, 13} defined as:

h = {(2, 7), (3, 9), (4, 11), (5, 13)}

It is seen that all distinct elements of the set {2, 3, 4, 5} have distinct images under h.

∴Function h is one-one.

Also, h is onto since for every element y of the set {7, 9, 11, 13}, there exists an element x in the set {2, 3, 4, 5}such that h(x) = y.

Thus, h is a one-one and onto function. Hence, h has an inverse.

Comments

Show that f: [−1, 1] → R, given byis one-one. Find the inverse of the function f: [−1, 1] → Range f.

(Hint: For y ∈Range fy =, for some x in [−1, 1], i.e.,)

f: [−1, 1] → R is given as

Let f(x) = f(y).

f is a one-one function.

It is clear that f: [−1, 1] → Range f is onto.

f: [−1, 1] → Range f is one-one and onto and therefore, the inverse of the function:

f: [−1, 1] → Range f exists.

Let g: Range f → [−1, 1] be the inverse of f.

Let y be an arbitrary element of range f.

Since f: [−1, 1] → Range f is onto, we have:

Now, let us define g: Range f → [−1, 1] as

gof = and fog =

f−1 = g

Comments

Consider fR → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.

f: RR is given by,

f(x) = 4x + 3

One-one:

Let f(x) = f(y).

f is a one-one function.

Onto:

For yR, let y = 4x + 3.

Therefore, for any y R, there exists such that

f is onto.

Thus, f is one-one and onto and therefore, f−1 exists.

Let us define g: RR by.

Hence, f is invertible and the inverse of f is given by

Comments

Consider fR→ [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f−1 of given by, where R+ is the set of all non-negative real numbers.

f: R+ → [4, ∞) is given as f(x) = x2 + 4.

One-one:

Let f(x) = f(y).

f is a one-one function.

Onto:

For y ∈ [4, ∞), let y = x2 + 4.

Therefore, for any y R, there exists such that

.

f is onto.

Thus, f is one-one and onto and therefore, f−1 exists.

Let us define g: [4, ∞) → R+ by,

Hence, f is invertible and the inverse of f is given by

Comments

Consider fR+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with.

f: R+ → [−5, ∞) is given as f(x) = 9x2 + 6x − 5.

Let y be an arbitrary element of [−5, ∞).

Let y = 9x2 + 6x − 5.

f is onto, thereby range f = [−5, ∞).

Let us define g: [−5, ∞) → R+ as

We now have:

and

Hence, f is invertible and the inverse of f is given by

Comments

Let fX → Y be an invertible function. Show that f has unique inverse.

(Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y,

fog1(y) = IY(y) = fog2(y). Use one-one ness of f).

Let f: XY be an invertible function.

Also, suppose f has two inverses (say).

Then, for all yY, we have:

Hence, f has a unique inverse.

Comments

Consider f: {1, 2, 3} → {abc} given by f(1) = af(2) = b and f(3) = c. Find f−1 and show that (f−1)−1 = f.

Function f: {1, 2, 3} → {a, b, c} is given by,

f(1) = a, f(2) = b, and f(3) = c

If we define g: {a, b, c} → {1, 2, 3} as g(a) = 1, g(b) = 2, g(c) = 3, then we have:

and, where X = {1, 2, 3} and Y= {a, b, c}.

Thus, the inverse of f exists and f−1= g.

f−1: {a, b, c} → {1, 2, 3} is given by,

f−1(a) = 1, f−1(b) = 2, f-1(c) = 3

Let us now find the inverse of f−1 i.e., find the inverse of g.

If we define h: {1, 2, 3} → {a, b, c} as

h(1) = a, h(2) = b, h(3) = c, then we have:

, where X = {1, 2, 3} and Y = {a, b, c}.

Thus, the inverse of g exists and g−1 = h ⇒ (f−1)−1 = h.

It can be noted that h = f.

Hence, (f−1)−1 = f.

Comments

Let fX → Y be an invertible function. Show that the inverse of f−1 is f, i.e.,

(f−1)−1 = f.

Let f: XY be an invertible function.

Then, there exists a function g: YX such that gof = IXand fog = IY.

Here, f−1 = g.

Now, gof = IXand fog = IY

f−1of = IXand fof−1= IY

Hence, f−1: YX is invertible and f is the inverse of f−1

i.e., (f−1)−1 = f.

Comments

If f→ be given by, then fof(x) is

(A) (B) x3 (C) x (D) (3 − x3)

f: RR is given as.

The correct answer is C.

Comments

Letbe a function defined as. The inverse of f is map g: Range

(A) (B)

(C) (D)

It is given that

Let y be an arbitrary element of Range f.

Then, there exists x ∈such that 

Let us define g: Rangeas

Now,

Thus, g is the inverse of f i.e., f−1 = g.

Hence, the inverse of f is the map g: Range, which is given by

The correct answer is B.

Comments

How helpful was it?

How can we Improve it?

Please tell us how it changed your life *

Please enter your feedback

Please enter your question below and we will send it to our tutor communities to answer it *

Please enter your question

Please select your tags

Please select a tag

Name *

Enter a valid name.

Email *

Enter a valid email.

Email or Mobile Number: *

Please enter your email or mobile number

Sorry, this phone number is not verified, Please login with your email Id.

Password: *

Please enter your password

By Signing Up, you agree to our Terms of Use & Privacy Policy

Thanks for your feedback

About UrbanPro

UrbanPro.com helps you to connect with the best Class 12 Tuition in India. Post Your Requirement today and get connected.

X

Looking for Class 12 Tuition Classes?

Find best tutors for Class 12 Tuition Classes by posting a requirement.

  • Post a learning requirement
  • Get customized responses
  • Compare and select the best

Looking for Class 12 Tuition Classes?

Get started now, by booking a Free Demo Class

This website uses cookies

We use cookies to improve user experience. Choose what cookies you allow us to use. You can read more about our Cookie Policy in our Privacy Policy

Accept All
Decline All

UrbanPro.com is India's largest network of most trusted tutors and institutes. Over 55 lakh students rely on UrbanPro.com, to fulfill their learning requirements across 1,000+ categories. Using UrbanPro.com, parents, and students can compare multiple Tutors and Institutes and choose the one that best suits their requirements. More than 7.5 lakh verified Tutors and Institutes are helping millions of students every day and growing their tutoring business on UrbanPro.com. Whether you are looking for a tutor to learn mathematics, a German language trainer to brush up your German language skills or an institute to upgrade your IT skills, we have got the best selection of Tutors and Training Institutes for you. Read more