Relations and Functions (Part – 2)

Questions Discussed:

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

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

Question 3. Let \text{A = \{1, 2, 3\}, B = \{4, 5, 6, 7\}} and let f = \text{ \{(1, 4), (2, 5), (3, 6)\}} be a function from A to B. Show that f is one-one. (N)

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

Question 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}}_{*}}?(N)

Questions Discussed:

Question 6. the injectivity and surjectivity of the following functions: (N)
  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}}

Question 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. (N)

Question 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. (N)

Question 9.Show that the Signum Function f:\mathbf{R}\to \mathbf{R}, given by {\displaystyle 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. (N)

Question 10.In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (N)
  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}}

Question 11. Let f:\mathbf{R}\to \mathbf{R} be defined as f(x)={{x}^{4}}. Choose the correct answer. (N)
  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.

Question 12.Let f:\mathbf{R}\to \mathbf{R} be defined as f(x)=3x. Choose the correct answer.(N)
  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.

Questions Discussed:

Question 1.If f:\mathbf{R}\to \text{A}, given by f(x)={{x}^{2}}-2x+2 is onto function, find set A. (B)

Question 2.Is f:\mathbf{R}\to \mathbf{R}, given by f(x)=\,|x-1| one-one? Give reason. (B)

Question 3. f:\mathbf{R}\to B given by f(x)=\sin x is onto function, then write the set B. (B)

Question 4.Let A = \mathbf{R} - \{3\} \text{ and } B = \mathbf{R} - \{1\}. Consider the function f:\mathbf{A}\to \mathbf{B} defined by f(x)={\displaystyle \left( {\frac{{x-2}}{{x-3}}} \right)}. Is f one-one and onto? Justify your answer. (N)

Question 5.Check the following functions for one-one and onto: (B)
 a. f:\mathbf{R}\to \mathbf{R},\,\,f(x) = {\displaystyle \frac{{2x-3}}{7} }
 b. f:\mathbf{R}\to \mathbf{R},\,\,f(x)=\,|x+1|
 c. f:\mathbf{R}-\{2\}\to \mathbf{R},\,\,f(x)= {\displaystyle \frac{{3x-1}}{{x-2}} }

Question 6.Show that the function f:\mathbf{R}\to \mathbf{R} defined by f(x)= {\displaystyle \frac{x}{{{{x}^{2}}+1}} },\,\forall \,x\in \mathbf{R}, is neither one-one nor onto. (E)

Questions Discussed:

Question 7. Let A=[-1,\,\,1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective: (E)
 a. f(x)={\displaystyle \frac{x}{2} }
 b. g(x)=\,\,|x|
 c. h(x)=\,\,x|x|
 d. k(x)={{x}^{2}}

Question 8. Let \displaystyle f:\mathbf{N}\to \mathbf{N} be defined by \text{for}\,\,\text{all}\,\,n\in \mathbf{N}. State whether the function f is bijective. Justify your answer. (N)

Question 9. Let A and B be sets. Show that f : A \times B → B \times A that f (a, b) = (b, a) is bijective function.

Questions Discussed:

Question 1. Let f :\{1, 3, 4\} \rightarrow \{1, 2, 5\} \text{ and } g:\{1, 2, 5\} \rightarrow \{1, 3\} be given by f = \{(1, 2), (3, 5), (4, 1)\} \text{ and } g = \{(1, 3), (2, 3), (5, 1)\} Write down gof. (N)

Question 2. If f = \{(5, 2), (6, 3)\}, g = \{(2, 5), (3, 6)\}, write f o g. (E)

Question 3. If the mappings f and g are given by f = \{(1, 2), (3, 5), (4, 1)\} \text{ and } g = \{(2, 3), (5, 1), (1, 3)\}, write f o g. (E)

Question 4. If f : \{1, 3\} \rightarrow \{1, 2, 5\} \text{ and } g: \{1, 2, 5\} \rightarrow \{1, 2, 3, 4\} be given by f = \{(1, 2), (3, 5)\}, g = \{( 1, 3), (2, 3), (5, 1 )\} , write gof . (B)

Question 5. Consider f : \mathbf{N} \rightarrow \mathbf{N}, g: \mathbf{N} \rightarrow \mathbf{N} \text{ and } h: \mathbf{N} \rightarrow \mathbf{R} defined as f (x) = 2x, g(y) = 3y + 4 \text{ and } h(z) = \sin z, \forall x, y \text{ and } z \text{ in } \mathbf{N}. Show that ho(gof) = (hog)of. (N)

Question 6. Find gof and fog, if (N)
a. f(x)=\,\,|x|\,\,\text{and }g(x)=\,\,|5x-2|
b. f(x)=8{{x}^{3}}\text{ and }g(x)={\displaystyle {{x}^{{\frac{1}{3}}}}}

Questions Discussed:

Question 7. If f : \mathbf{R} \rightarrow \mathbf{R} be given by f(x)={\displaystyle{{(3-{{x}^{3}})}^{{\frac{1}{3}}}}}, then find fof(x). (N)

Question 8. Let g,\,\,f:\mathbf{R}\to \mathbf{R} be defined by f(x)=2x+1 and g(x)={{x}^{2}}-2,\,\,\forall \,\,x\in \mathbf{R}, respectively. Then, find g\,o\,f. (E)

Question 9. Let g,\,\,f:\mathbf{R}\to \mathbf{R} be defined by g(x)={\displaystyle \frac{{x+2}}{3},\,\,f(x) }=3x-2 write fog(x). (B)

Question 10. If f : \mathbf{R} \rightarrow \mathbf{R} defined by f(x)={\displaystyle \frac{{x-1}}{2}}, find (fof)(x). (B)

Question 11. If f : \mathbf{R} \rightarrow \mathbf{R} defined by f(x)={{x}^{2}}-3x+2, find (fof)(x). (E)

Question 12. If f(x)={\displaystyle \log \left( {\frac{{1+x}}{{1-x}}} \right)}, show that f {\displaystyle \left( {\frac{{2x}}{{1+{{x}^{2}}}}} \right) } =2f(x). (B)

Question 13. Let f:\mathbf{R}\to \mathbf{R} be defined by f(x)={{x}^{2}}+1 , find the pre image of 17 and -3. (B)

Question 14. If f:\mathbf{R} \to \mathbf{R},\,\,g:\mathbf{R}\to \mathbf{R}, given by f(x)=[x],\,\,g(x)=\,|x|, then find fog{\displaystyle \left( {-\frac{2}{3}} \right)} and gof {\displaystyle\left( {-\frac{2}{3}} \right)}. (B)

Questions Discussed:

Question 1. Let S = \{1, 2, 3\}. Determine whether the functions f : S \to S defined as below have inverses. Find {{f}^{{-1}}}, if it exists. (N)
a. f = \{(1, 1), (2, 2), (3, 3)\}
b. f = \{(1, 2), (2, 1), (3, 1)\}
c. f = \{(1, 3), (3, 2), (2, 1)\}

Question 2. State with reason whether following functions have inverse (N)
a. f : \{1, 2, 3, 4\} \to \{10\} with f = \{(1, 10), (2, 10), (3, 10), (4, 10)\}
b. g : \{5, 6, 7, 8\} \to \{1, 2, 3, 4\} with g = \{(5, 4), (6, 3), (7, 4), (8, 2)\}
c. h : \{2, 3, 4, 5\} \to \{7, 9, 11, 13\} with h = \{(2, 7), (3, 9), (4, 11), (5, 13)\}

Questions Discussed:

Question 4. Let f : \mathbf{N} \to Y be a function defined as f (x) = 4x + 3, where, Y = \{y \in \mathbf{N}: y = 4x + 3\text{ for some } x \in \mathbf{N}\}. Show that f is invertible. Find the inverse. (N)

Question 5. Let \text{Y}=\{{{n}^{2}}:n\in \mathbf{N}\}\,\subset \,\mathbf{N}. Consider f:\mathbf{N}\to \text{Y} as f(n)={{n}^{2}}. Show that f is invertible. Find the inverse of f. (N)

Question 6. If f(x)={\displaystyle \frac{{4x+3}}{{6x-4}},\,x\ne \frac{2}{3},\,} show that fof(x)=x, for all x\ne {\displaystyle \frac{2}{3} }. What is the inverse of f ? (N) (Involution or Involutory Function)

Question 7. Show that f:[-1,\,\,1]\to \mathbf{R}, given by f(x)={\displaystyle \frac{x}{{x+2}} } is one-one. Find the inverse of the function f:[-1,\,\,1]\to \text{Range}\,\,f. (N)

Questions Discussed:

Question 3. Let f : X \to Y be an invertible function. Show that f has unique inverse. (N)

Question 8. Let f:\mathbf{R}-{\displaystyle \left\{ {-\frac{4}{3}} \right\} }\to \mathbf{R} be a function defined as f(x)={\displaystyle \frac{{4x}}{{3x+4}} }. The inverse of f is the map g:\text{Range }f\to \mathbf{R}- {\displaystyle \left\{ {-\frac{4}{3}} \right\} } given by (N)
 a. g(y)={\displaystyle \frac{{3y}}{{3-4y}}}
 b. \displaystyle g(y)=\frac{{4y}}{{4-3y}}
 c. g(y)={\displaystyle \frac{{4y}}{{3-4y}} }
 d. g(y)={\displaystyle \frac{{3y}}{{4-3y}} }

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

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

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

Questions Discussed:

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

Question 5. Consider f : \{1, 2, 3\} \to \{a, b, c\} given by f (1) = a, f (2) = b \text{ and } f (3) = c. Find {{f}^{{-1}}} and show that {{({{f}^{{-1}}})}^{{-1}}}=f. (N)

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

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

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