ashish kumar

Ch1. Relations and Functions (Part – 2)

In this self study course, you will learn One to one and onto functions, composite functions and inverse of a function. For further understanding of concepts and for examination preparation, practice questions based on the above topics are discussed in the form of assignments that have questions from NCERT Textbook exercise, NCERT Examples, Board’s Question Bank, RD Sharma, NCERT Exemplar etc. instead of only one book. The PDF of assignments can be downloaded within the course.

Total Video Duration Β 8 hr 22 min 00 sec
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Assignment – 1

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Question – 1

1. Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subsets of X Γ—Y are functions from X to Y or not.
a. f = {(1, 4), (1, 5), (2, 4), (3, 5)}
b. h = {(1,4), (2, 5), (3, 5)}
c. g = {(1, 4), (2, 4), (3, 4)}
d. k = {(1,4), (2, 5)}.

 05:21

Question – 2

2. If f:A\to B is bijective function such that n(A) =10, then n(B) = ?

 01:56

Question – 4

4. Let A be the set of all 50 students of Class X in a school. Let \displaystyle f:A\to N be function defined by f(x) = roll number of the student x. Show that f is one-one but not onto.

 04:47

Question – 5

5. Show that the function f:{{\mathbf{R}}_{*}}\to {{\mathbf{R}}_{*}} defined by f(x)=\frac{1}{x} is one-one and onto, where {{\mathbf{R}}_{*}} is the set of all non-zero real numbers. Is the result true, if the domain {{\mathbf{R}}_{*}} is replaced by \mathbf{N} with co-domain being same as {{\mathbf{R}}_{*}}?

 09:32

Question – 6

6. Check the injectivity and surjectivity of the following functions:
a. f:\mathbf{N}\to \mathbf{N} given by f(x)={{x}^{2}}
b. f:\mathbf{Z}\to \mathbf{Z} given by f(x)={{x}^{2}}
c. f:\mathbf{R}\to \mathbf{R} given by f(x)={{x}^{2}}
d. f:\mathbf{N}\to \mathbf{N} given by f(x)={{x}^{3}}
e. f:\mathbf{Z}\to \mathbf{Z} given by f(x)={{x}^{3}}

 25:29

Question – 7

7. Prove that the Greatest Integer Function f:\mathbf{R}\to \mathbf{R}, given by f(x)=[x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

 06:33

Question – 8

8. Show that the Modulus Function f:\mathbf{R}\to \mathbf{R}, given by f(x)=\,|x|, is neither one-one nor onto, where | x | is x, if x is positive or 0 and | x | is – x, if x is negative.

 04:34

Question – 9

9. Show that the Signum Function f:\mathbf{R}\to \mathbf{R}, given by f(x)=\begin{cases} {1}, & \text{ if } x > 0 \\ {0}, & \text{ if } x = 0 \\ {-1}, & \text{ if } x < 0 \end{cases} is neither one-one nor onto.

 04:20

Question – 10

10. In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
a. f:\mathbf{R}\to \mathbf{R} defined by f(x)=3-4x
b. f:\mathbf{R}\to \mathbf{R} defined by f(x)=1+{{x}^{2}}

 08:46

Question – 11, 12

11. Let f:\mathbf{R}\to \mathbf{R} be defined as f(x)={{x}^{4}}. Choose the correct answer.
a. f is one-one onto
b. f is one-one but not onto
c. f is many-one onto
d. f is neither one-one nor onto.

12. Let f:\mathbf{R}\to \mathbf{R} be defined as f(x)=3x. Choose the correct answer.
a. f is one-one onto
b. f is one-one but not onto
c. f is many-one onto
d. f is neither one-one nor onto.

 06:49

Assignment – 2

9 Subtopics
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Assignment – 4

6 Subtopics
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Assignment – 5

8 Subtopics
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Question – 1

1. Consider { \displaystyle f:{{\mathbf{R}}_{+}}\to [4,\,\,\infty )} given by { \displaystyle f(x)={{x}^{2}}+4} . Show that { \displaystyle f} is invertible with the inverse { \displaystyle {{f}^{{-1}}}} of { \displaystyle f} given by { \displaystyle {{f}^{{-1}}}(y)=\sqrt{{y-4}}} , where { \displaystyle {{\mathbf{R}}_{+}}} is the set of all non-negative real numbers.

 11:52

Question – 2

2. Consider { \displaystyle f:{{\mathbf{R}}_{+}}\to [-5,\,\,\infty )} given by { \displaystyle f(x)=9{{x}^{2}}+6x-5} . Show that { \displaystyle f} is invertible with { \displaystyle {{f}^{{-1}}}(y)=\left( {\frac{{\sqrt{{y+6}}-1}}{3}} \right)} .

 15:38

Question – 3

3. Let { \displaystyle f':\mathbf{N}\to \mathbf{R}} be a function defined as { \displaystyle f'(x)=4{{x}^{2}}+12x+15} . Show that { \displaystyle f:\mathbf{N}\to \text{S}} , where, S is the range of f, is invertible. Find the inverse of { \displaystyle f} .

 10:53

Question – 4

4. If the function { \displaystyle f:\mathbf{R}\to \mathbf{R}} be defined by { \displaystyle f(x)=2x-3} and { \displaystyle g:\mathbf{R}\to \mathbf{R}} by { \displaystyle g(x)={{x}^{3}}+5} , then show that { \displaystyle fog} is invertible. Also find { \displaystyle {{(fog)}^{{-1}}}(x)} , hence find { \displaystyle {{(fog)}^{{-1}}}(9)} .

 14:28

Question – 5

5. Consider f : {1, 2, 3} β†’ {a, b, c} given by f (1) = a, f (2) = b and f (3) = c. Find { \displaystyle {{f}^{{-1}}}} and show that { \displaystyle {{({{f}^{{-1}}})}^{{-1}}}=f} .

 02:59

Question – 7

7. Let f: X β†’ Y be an invertible function. Show that the inverse of { \displaystyle {{f}^{{-1}}}} is { \displaystyle f} , i.e., { \displaystyle {{({{f}^{{-1}}})}^{{-1}}}=f} .

 12:01

Question – 6

6. Let f, g and h be functions from R to R. Show that
a. { \displaystyle (f+g)oh=foh+goh}
b. { \displaystyle (f.g)oh=(foh).(goh)}

07:00

Question – 8

8. Consider f : {1, 2, 3} β†’ {a, b, c} and g : {a, b, c} β†’ {apple, ball, cat} defined as f(1) = a, f(2) = b, f(3) = c, g(a) = apple, g(b) = ball and g(c) = cat. Show that f, g and gof are invertible. Find out { \displaystyle {{f}^{{-1}}},\,\,{{g}^{{-1}}}\,\text{and }{{(gof)}^{{-1}}}} and show that { \displaystyle {{(gof)}^{{-1}}}={{f}^{{-1}}}o{{g}^{{-1}}}} .

 13:09