# Ch01. Relations and Functions (Part – 2)

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In this online 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.

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The following list of questions are just meant for reference before purchasing membership. The list might or might not include NCERT Questions as it depends on the chapter/course. Some chapters have NCERT questions combined in the Assignments and some chapters have separate NCERT questions and Assignments. For complete details, please check the index of the course in the "About Course".

### Assignment – 1

1. Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subsets of X ×Y are functions from X to Y or not.
1. f = {(1, 4), (1, 5), (2, 4), (3, 5)}
2. h = {(1,4), (2, 5), (3, 5)}
3. g = {(1, 4), (2, 4), (3, 4)}
4. k = {(1,4), (2, 5)}.
2. If {\displaystyle f:A \to B} is bijective function such that n(A) =10, then n(B) = ? (B)
3. 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. Show that f is one-one. (N)
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. (N)
5. Show that the function {\displaystyle f:{{\bf{R}}_*} \to {{\bf{R}}_*}} defined by {\displaystyle f(x) = \frac{1}{x}} is one-one and onto, where {\displaystyle {{\bf{R}}_*}} is the set of all non-zero real numbers. Is the result true, if the domain {\displaystyle {{\bf{R}}_*}} is replaced by {\displaystyle {\bf{N}}} with co-domain being same as {\displaystyle {{\bf{R}}_*}} ?(N)
6. Check the injectivity and surjectivity of the following functions: (N)
1. {\displaystyle f:{\bf{N}} \to {\bf{N}}} given by {\displaystyle f(x) = {x^2}}
2. {\displaystyle f:{\bf{Z}} \to {\bf{Z}}} given by {\displaystyle f(x) = {x^2}}
3. {\displaystyle f:{\bf{R}} \to {\bf{R}}} given by {\displaystyle f(x) = {x^2}}
4. {\displaystyle f:{\bf{N}} \to {\bf{N}}} given by {\displaystyle f(x) = {x^3}}
5. {\displaystyle f:{\bf{Z}} \to {\bf{Z}}} given by {\displaystyle f(x) = {x^3}}
7. Prove that the Greatest Integer Function {\displaystyle f:{\bf{R}} \to {\bf{R}}} , given by {\displaystyle f(x) = [x]} , is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x. (N)
8. Show that the Modulus Function {\displaystyle f:{\bf{R}} \to {\bf{R}}} , given by {\displaystyle 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. (N)
9. Show that the Signum Function {\displaystyle f:{\bf{R}} \to {\bf{R}}} , given by {\displaystyle f(x) = \left\{ \begin{array}{l} 1, & {\rm{if}}\,x > 0\\ 0, & {\rm{if}}\,x = 0\\ - 1, & {\rm{if}}\,x < 0\end{array} \right.} is neither one-one nor onto. (N)
10. In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (N)
1. {\displaystyle f:{\bf{R}} \to {\bf{R}}} defined by {\displaystyle f(x) = 3 - 4x}
2. {\displaystyle f:{\bf{R}} \to {\bf{R}}} defined by {\displaystyle f(x) = 1 + {x^2}}
11. Let {\displaystyle f:{\bf{R}} \to {\bf{R}}} be defined as {\displaystyle f(x) = {x^4}} . Choose the correct answer. (N)
1. f is one-one onto
2. f is one-one but not onto
3. f is many-one onto
4. f is neither one-one nor onto.
12. Let {\displaystyle f:{\bf{R}} \to {\bf{R}}} be defined as {\displaystyle f(x) = 3x} . Choose the correct answer. (N)
1. f is one-one onto
2. f is one-one but not onto
3. f is many-one onto
4. f is neither one-one nor onto.

### Assignment – 2

1. If {\displaystyle f:{\bf{R}} \to {\rm{A}}} , given by {\displaystyle f(x) = {x^2} - 2x + 2} is onto function, find set A. (B)
2. Is {\displaystyle f:{\bf{R}} \to {\bf{R}}} , given by {\displaystyle f(x) = \,|x - 1|} one-one? Give reason. (B)
3. {\displaystyle f:{\bf{R}} \to B} given by {\displaystyle f(x) = \sin x} is onto function, then write the set B. (B)
4. Let A = R – {3} and B = R – {1}. Consider the function {\displaystyle f:{\bf{A}} \to {\bf{B}}} defined by {\displaystyle f(x) = \left( {\frac{{x - 2}}{{x - 3}}} \right)} . Is f one-one and onto? Justify your answer. (N)
5. Check the following functions for one-one and onto: (B)
1. {\displaystyle f:{\bf{R}} \to {\bf{R}},\,\,f(x) = \frac{{2x - 3}}{7}}
2. {\displaystyle f:{\bf{R}} \to {\bf{R}},\,\,f(x) = \,|x + 1|}
3. {\displaystyle f:{\bf{R}} - \{ 2\} \to {\bf{R}},\,\,f(x) = \frac{{3x - 1}}{{x - 2}}}
6. Show that the function {\displaystyle f:{\bf{R}} \to {\bf{R}}} defined by {\displaystyle f(x) = \frac{x}{{{x^2} + 1}},\,\forall \,x \in {\bf{R}}} , is neither one-one nor onto. (E)
7. Let {\displaystyle A = [ - 1,\,\,1]} . Then, discuss whether the following functions defined on A are one-one, onto or bijective: (E)
1. {\displaystyle f(x) = \frac{x}{2}}
2. {\displaystyle g(x) = \,\,|x|}
3. {\displaystyle h(x) = \,\,x|x|}
4. {\displaystyle k(x) = {x^2}}
8. Let {\displaystyle f:{\bf{N}} \to {\bf{N}}} be defined by {\displaystyle f(n) = \left\{ \begin{array}{l} \frac{{n + 1}}{2},\, & {\rm{if}}\,n\,\,{\rm{is}}\,\,{\rm{odd}}\\ \frac{n}{2},\, & {\rm{if}}\,n\,\,{\rm{is}}\,\,{\rm{even}} \end{array} \right.\,} {\displaystyle {\rm{for}}\,\,{\rm{all}}\,\,n \in {\bf{N}}} . State whether the function f is bijective. Justify your answer. (N)
9. Let A and B be sets. Show that f : A × B → B × A such that f (a, b) = (b, a) is bijective function.

### Assignment – 3

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. (N)
2. If f = {(5, 2), (6, 3)}, g = {(2, 5), (3, 6)}, write f o g. (E)
3. If the mappings f and g are given by f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, write f o g. (E)
4. If f : {1, 3} →{1, 2, 5} and g: {1, 2, 5} →{1, 2, 3, 4} be given by f = {(1, 2), (3, 5)}, g = {( 1, 3), (2, 3), (5, 1 )}, write {\displaystyle gof} . (B)
5. Consider f : NN, g: NN and h: NR defined as f (x) = 2x, g(y) = 3y + 4 and h(z) = sin z, ∀ x, y and z in N. Show that ho(gof) = (hog)of. (N)
6. Find {\displaystyle gof} and {\displaystyle fog} , if (N)
1. {\displaystyle f(x) = \,\,|x|\,\,{\rm{and }}g(x) = \,\,|5x - 2|}
2. {\displaystyle f(x) = 8{x^3}{\rm{ and  }}g(x) = {x^{\frac{1}{3}}}}
7. If f : RR be given by {\displaystyle f(x) = {(3 - {x^3})^{\frac{1}{3}}}} , then find {\displaystyle fof(x)} . (N)
8. Let {\displaystyle g,\,\,f:{\bf{R}} \to {\bf{R}}} be defined by {\displaystyle f(x) = 2x + 1} and {\displaystyle g(x) = {x^2} - 2,\,\,\forall \,\,x \in {\bf{R}}} , respectively. Then, find {\displaystyle g\,o\,f} . (E)
9. Let {\displaystyle g,\,\,f:{\bf{R}} \to {\bf{R}}} be defined by {\displaystyle g(x) = \frac{{x + 2}}{3},\,\,f(x) = 3x - 2} write {\displaystyle fog(x)} . (B)
10. If f : RR defined by {\displaystyle f(x) = \frac{{x - 1}}{2}} , find {\displaystyle (fof)(x)} . (B)
11. If f : RR defined by {\displaystyle f(x) = {x^2} - 3x + 2} , find {\displaystyle (fof)(x)} . (E)
12. If {\displaystyle f(x) = \log \left( {\frac{{1 + x}}{{1 - x}}} \right)} , show that {\displaystyle f\left( {\frac{{2x}}{{1 + {x^2}}}} \right) = 2f(x)} . (B)
13. Let {\displaystyle f:{\bf{R}} \to {\bf{R}}} be defined by {\displaystyle f(x) = {x^2} + 1} , find the pre image of 17 and -3. (B)
14. If {\displaystyle f:{\bf{R}} \to {\bf{R}},\,\,g:{\bf{R}} \to {\bf{R}}} , given by {\displaystyle f(x) = [x],\,\,g(x) = \,|x|} , then find {\displaystyle fog\left( { - \frac{2}{3}} \right)} and {\displaystyle gof\left( { - \frac{2}{3}} \right)} . (B)

### Assignment – 4

1. Let S = {1, 2, 3}. Determine whether the functions f : S → S defined as below have inverses. Find {\displaystyle {f^{ - 1}}} , if it exists. (N)
1. f = {(1, 1), (2, 2), (3, 3)}
2. f = {(1, 2), (2, 1), (3, 1)}
3. f = {(1, 3), (3, 2), (2, 1)}
2. State with reason whether following functions have inverse (N)
1. f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
2. g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}
3. h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}
3. Let f : X → Y be an invertible function. Show that f has unique inverse. (N)
4. Let f : N → Y be a function defined as f (x) = 4x + 3, where, Y = {yN: y = 4x + 3 for some xN}. Show that f is invertible. Find the inverse. (N)
5. Let {\displaystyle {\rm{Y}} = \{ {n^2}:n \in {\bf{N}}\} \, \subset \,{\bf{N}}.} Consider {\displaystyle f:{\bf{N}} \to {\rm{Y}}} as {\displaystyle f(n) = {n^2}} . Show that f  is invertible. Find the inverse of  f. (N)
6. If {\displaystyle f(x) = \frac{{4x + 3}}{{6x - 4}},\,x \ne \frac{2}{3},\,} show that {\displaystyle fof(x) = x} , for all {\displaystyle x \ne \frac{2}{3}} . What is the inverse of ? (N) (Involution or Involutory Function)
7. Show that {\displaystyle f:[ - 1,\,\,1] \to {\bf{R}}} , given by {\displaystyle f(x) = \frac{x}{{x + 2}}} is one-one. Find the inverse of the function {\displaystyle f:[ - 1,\,\,1] \to {\rm{Range}}\,\,f} . (N)
8. Let {\displaystyle f:{\bf{R}} - \left\{ { - \frac{4}{3}} \right\} \to {\bf{R}}} be a function defined as {\displaystyle f(x) = \frac{{4x}}{{3x + 4}}} . The inverse of {\displaystyle f} is the map {\displaystyle g:{\rm{Range }}f \to {\bf{R}} - \left\{ { - \frac{4}{3}} \right\}} given by (N)
1. {\displaystyle g(y) = \frac{{3y}}{{3 - 4y}}}
2. {\displaystyle g(y) = \frac{{4y}}{{4 - 3y}}}
3. {\displaystyle g(y) = \frac{{4y}}{{3 - 4y}}}
4. {\displaystyle g(y) = \frac{{3y}}{{4 - 3y}}}

### Assignment – 5

1. Consider {\displaystyle f:{{\bf{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 {{\bf{R}}_ + }} is the set of all non-negative real numbers. (N)
2. Consider {\displaystyle f:{{\bf{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)} . (N)
3. Let {\displaystyle f':{\bf{N}} \to {\bf{R}}} be a function defined as {\displaystyle f'(x) = 4{x^2} + 12x + 15} . Show that {\displaystyle f:{\bf{N}} \to {\rm{S}}} , where, S is the range of f, is invertible. Find the inverse of {\displaystyle f} . (N)
4. If the function {\displaystyle f:{\bf{R}} \to {\bf{R}}} be defined by {\displaystyle f(x) = 2x - 3} and {\displaystyle g:{\bf{R}} \to {\bf{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)} . (B)
5. Consider f : {1, 2, 3} → {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} . (N)
6. Let f, g and h be functions from R to R. Show that (N)
1. {\displaystyle (f + g)oh = foh + goh}
2. {\displaystyle (f.g)oh = (foh).(goh)}
7. Let f: X → Y be an invertible function. Show that the inverse of {\displaystyle {f^{ - 1}}} is {\displaystyle f} , i.e., {\displaystyle {({f^{ - 1}})^{ - 1}} = f} . (N)
8. Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f(1) = a, f(2) = bf(3) = cg(a) = apple,  g(b) = ball and  g(c) = cat. Show that f, g and gof are invertible. Find out {\displaystyle {f^{ - 1}},\,\,{g^{ - 1}}\,{\rm{and }}{(gof)^{ - 1}}} and show that {\displaystyle {(gof)^{ - 1}} = {f^{ - 1}}o{g^{ - 1}}} . (N)
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