NCERT Solutions for Class 12 Maths Chapter 1 Excercise 1.3 (Relations and Functions)
NCERT Solutions for Class 12 Maths Chapter 1 Excercise 1.3: Get accurate error free NCERT Solutions for Class 12 Maths chapter 1-Relations and Functions Exercise 1.3.
Krati Saraswat12 Jan, 2024
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NCERT Solutions for Class 12 Maths Chapter 1 Excercise 1.3 (Relations and Functions)
NCERT Solutions for Class 12 Maths Chapter 1 Excercise 1.3 of Relations and Functions is prepared by academic team of Physics Wallah. We have prepared NCERT Solutions for all exercise of chapter 1. Given below is step by step solutions of all questions given in NCERT textbook for Chapter 1 Relations and Functions.NCERT Solutions for Class 12 Physics
NCERT Solutions for Class 12 Maths Chapter 1 Excercise 1.3 Overview
NCERT Solutions for Class 12 Maths Chapter 1 Excercise 1.3 addresses these significant subjects. In order to fully comprehend the concepts presented in the chapter and make effective use of the provided solutions, it is recommended that students go over each topic in great detail. The instructors at Physics Wallah have put a lot of effort into helping students better understand the ideas presented in this chapter, as evidenced by these solutions. The intention is for students to effortlessly achieve excellent exam scores after reviewing and practicing these responses.NCERT Solutions for Class 12 Maths Chapter 1 Excercise 1.3 PDF
Educators at Physics Wallah has created NCERT Solutions for Class 12 Maths Chapter 1 Excercise 1.3 comprehensive solutions for in order to assist students in comprehending and applying chapter themes. These questions are meant to help students understand explanations more easily. To download the NCERT Solutions for Class 12 Maths Chapter 1 Excercise 1.3 PDF, click the following link:NCERT Solutions Class 12 Maths Chapter 1 PDF Download Link
NCERT Solutions for Class 12 Maths Chapter 1 Excercise 1.3
Solve The Following Questions NCERT Solutions for Class 12 Maths Chapter 1 Excercise 1.3 (Relations and Functions)
Question 1. Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof. Solution :
NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.1
Question 2. Let f,g and h be functions from R to R. Show that: (f + g)oh = foh + goh (f.g)oh = (foh). (goh) Solution :
NCERT Solutions for Class 12 Maths Chapter 1 Miscellaneous Exercise
Question 3. Find gof and fog , if: (i) f(x) = |x| and g(x) = |5x - 2| (ii) f(x) = 8x 3 and g(x) = x 1/3 Solution :
Question
4. If
show that fof (x) = x for all x ≠ 2/3 What is the inverse of f
Solution :
Hence, the given function f is invertible and the inverse of f is itself.
NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.2
Question 5. 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)} Solution : (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.NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.4
Question 6. Show that f: [-1,1] → R given by f(x) = x/x+2 is one-one. Find the inverse of the function f: [-1,1] → Range f Solution :
Question
7. Consider f : R→ R given by f(x) = 4x + 3 Show that f is invertible. Find the inverse of f.
Solution :
Question
8. Consider
f
: R
+
→ [4, ∞) given by f(
x
) =
x
2
+ 4 Show that f is invertible with the inverse
f
−1
of f given by
f
−1
(y) = √y - 4 where R
+
is the set of all non-negative real numbers.
Solution :
Question
9. Consider
f
: R
+
→ [−5, ∞) given by
f
(
x
) = 9
x
2
+ 6
x
− 5. Show that
f
is invertible with
Solution :
Question
10. Let f: X → Y be an invertible function. Show that f has unique inverse.
(Hint: suppose g
1
and g
2
are two inverses of f. Then for all y ∈ Y,
fog
1
(y) = IY(y) = fog
2
(y). Use one-one ness of f).
Solution :
Question
11. Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f
−1
and show that (f
−1)−1
= f.
Solution :
Question
12. Let f: X → Y be an invertible function. Show that the inverse of f−1 is f, i.e., (f
−1
)
−1
= f.
Solution :
Let f: X → Y be an invertible function.
Then, there exists a function g: Y → X such that gof = IXand fog = IY.
Here, f
−1
= g.
Now, gof = IX and fog = IY
⇒ f
−1
of = IX and fof
−1
= IY
Hence, f
−1
: Y → X is invertible and f is the inverse of f
−1
i.e., (f−1)
−1
= f.
Question
13.
(A) 1/x
3
(
B)
x
3
(C) x
(D) (3 − x 3 )
Solution :
Question
14.
Solution :
NCERT Solutions for Class 12 Maths Chapter 1 FAQs
What is a relation function in 12th class?
Mathematically, “a relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.
How many exercises are there in relations and functions class 12?
Class 12 Maths Chapter 1 Relations and Functions has 55 questions in 4 exercises along with 19 extra questions provided in a miscellaneous exercise.
Which relation is called function?
A function is a relation which describes that there should be only one output for each input (or) we can say that a special kind of relation (a set of ordered pairs), which follows a rule i.e., every X-value should be associated with only one y-value is called a function.
What is the formula for relations?
It is written as R = X × Y where each element of X is related to every element of Y. Example, P = {3, 7, 9}, Q = {12, 18, 20} and R = {(x, y) where x < y}.
What is difference between function and relation?
A relation shows the relationship between input and output, and a function is a relation which derives one OUTPUT for each given INPUT.
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