Answer : Given: f(x) = x2 + 1 To find: (i) f-1{10} We know that, if f: X → Y such that y ∈ Y. Then f-1(y) = {x ∈ X: f(x) = y}. In other words, f-1(y) is the set of pre – images of y Let f-1{10} = x....
Answer : (i) f: R → R such that f(x) = x2 Since the value of x is squared, f(x) will always be equal or greater than 0. ∴ the range is [0, ∞) g: R → R such that g(x) = x2 + 1 Since, the value of x...
Answer : It is given that f : R → R and g : C → C Thus, Domain (f) = R and Domain (g) = C We know that, Real numbers ≠ Complex Number ∵, Domain (f) ≠ Domain (g) ∴ f(x) and g(x) are not equal...
Answer : Given that f: R → R such that f(x) = 2x To find: (i) Range of x Here, f(x) = 2x is a positive real number for every x ∈ R because 2x is positive for every x ∈ R. Moreover, for every...
Answer : Given that f: R+→ R such that f(x) = logex To find: (i) Range of f Here, f(x) = logex We know that the range of a function is the set of images of elements in the domain. ∴ The image set of...
Answer : Given: f(x) = highest prime factor of x And since x ∈ A, A = {12, 13, 14, 15, 16, 17} Value of x can only be 12, 13, 14, 15, 16, 17 Doing prime factorization of the above, we get Hence,...
Answer : Given: f(x) = x2 Firstly, we find the f(5) Putting the value of x = 5 in the given eq., we get f(5) = (5)2 ⇒ f(5) = 25 Similarly, f(1) = (1)2 ⇒ f(1) = 1 Putting the value of f(5) and f(1)...
Answer : Given that: f = {(0, -5), (1, -2), (3, 4), (4, 7)} be a function from Z to Z defined by linear function. We know that, linear functions are of the form y = mx + b Let f(x) = ax + b, for...
Answer : Given: g = {(1, 2), (2, 5), (3, 8), (4, 10), (5, 12), (6, 12)} We know that, A function ‘f’ from set A to set B is a correspondence (rule) which associates elements of set A to elements of...
Answer : Given: A = {2, 3, 5, 7} and B = {3, 5, 9, 13, 15} f = {(x, y): x ∈ A, y ∈ B and y = 2x – 1} For x = 2 y = 2x – 1 y = 2(2) – 1 y = 3 ∈ B For x = 3 y = 2x – 1 y = 2(3) – 1 y = 5 ∈ B For x = 5...
Answer : Given: A = {0, 1, 2} and B = {3, 5, 7, 9} f = {(x, y): x ∈ A, y ∈ B and y = 2x + 3} For x = 0 y = 2x + 3 y = 2(0) + 3 y = 3 ∈ B For x = 1 y = 2x + 3 y = 2(1) + 3 y = 5 ∈ B For x = 2 y = 2x...
Answer : Given: A = {1, 2} and B = {2, 4, 6} f = {(x, y): x ∈ A, y ∈ B and y > 2x + 1} Putting x = 1 in y > 2x + 1, we get y > 2(1) + 1 ⇒ y > 3 and y ∈ B this means y = 4, 6 if x = 1...
Answer : (i) Given: A = {-1, 0, 1, 2} and B = {2, 3, 4, 5} Function: all elements of the first set are associated with the elements of the second An element of the first set has a unique image in...
Answer : Given: f: X → R, f(x) = x3 + 1 Here, X = {-1, 0, 3, 7, 9} For x = -1 f(-1) = (-1)3 + 1 = -1 + 1 = 0 For x = 0 f(0) = (0)3 + 1 = 0 + 1 = 1 For x = 3 f(3) = (3)3 + 1 = 27 + 1 = 28 For x = 7...
Answer : X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16} and F = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)} To show: F is a relation from X to Y First elements in F = 1, 2, 3, 4 All the first elements...
Answer : Fundamental difference between Relation and Function: Every function is a relation, but every relation need not be a function. A relation f from A to B is called a function if Dom(f) = A no...
Answer : Function as a correspondence between two sets: Let A and B be two non – empty sets. Then, a function ‘f’ from set A to set B is a correspondence (rule) which associates elements of set A to...
Answer : Function as a set of ordered pairs: A function is a set of ordered pairs with the property that no two ordered pairs have the same first component and a different second component. The...
Solution: From the first eq., write $y$ in terms of $x$ $y=2-2 x\dots \dots(i)$ Substituting different values of $\mathrm{x}$ in equation(i) to get different values of $\mathrm{y}$ For $x=0,...
Solution: From the first equation, write y in terms of $x$ $y=6-2 x\dots \dots(i)$ Substituting different values of $x$ in equation(i) to get different values of $y$ For $x=0, y=6-0=6$ For $x=2,...
Solution: Draw a horizontal line on a graph paper X'OX and a vertical line YOY' as the x-axis and y-axis, respectively. Graph of $2 x+3 y=4$ $\begin{array}{l} 2 x+3 y=4 \\ \Rightarrow 3 y=(-2...
Solution: From the first eq., write y in terms of $x$ $\mathrm{y}=\frac{x-6}{2}\dots \dots(i)$ Substituting different values of $x$ in equation(i) to get different values of y For $x=-2,...
Solution: From the first eq., write y in terms of $x$ $y=\frac{x-5}{2}\dots \dots(i)$ Substituting different values of $x$ in eq.(i) to get different values of y For $x=-5, y=\frac{-5-5}{2}=-5$ For...
Solution: Draw a horizontal line on a graph paper $\mathrm{X}^{\prime} \mathrm{OX}$ and a vertical line YOY' representing the $\mathrm{x}-$ axis and $y$-axis, respectively. Graph of $2...
Solution: Draw a horizontal line on a graph paper X'OX and a vertical line YOY' representing the xaxis and y-axis, respectively. Graph of $3 x-y=5$ $3 x-y=5$ $\Rightarrow y=3 x-5\dots \dots(i)$...
Solution: From the first eq., write y in terms of $x$ $y=\frac{6-2 x}{3}\dots \dots(i)$ Substituting different values of $x$ in equation(i) to get different values of y For $x=-3, y=\frac{6+6}{3}=4$...
Solution: Draw a horizontal line on a graph paper $X^{\prime} O X$ and a vertical line YOY' as the $x$-axis and $y$-axis, respectively. $\text { Graph of } 4 x-3 y+4=0$ $\begin{array}{l} 4 x-3 y+4=0...
Solution: From the first eq., write y in terms of $x$ $y=\frac{2 x-12}{3}\dots \dots (i)$ Substituting different values of $\mathrm{x}$ in eq.(i) to get different values of $\mathrm{y}$ For $x=0,...
Solution: Draw a horizontal line on a graph paper $X^{\prime} O X$ and a vertical line YOY' as the $x$-axis and $y$-axis, respectively. $\begin{array}{l} \quad \text { Graph of } 2 x-5 y+4=0 \\ 2...
Solution: Draw a horizontal line on a graph paper $X^{\prime} O X$ and a vertical line YOY' as the $x$-axis and $y$-axis, respectively. $\text { Graph of } 2 x-y=1$ $\begin{array}{l} 2 x-y=1 \\...
Solution: Draw a horizontal line on a graph paper $X^{\prime} O X$ and a vertical line $Y O Y^{\prime}$ as the $x$-axis and $y$-axis, respectively. Graph of $4 x-y=4$ $\begin{array}{l} 4 x-y=4 \\...
Solution: Draw a horizontal line on a gfraph paper $X^{\prime} O X$ and a vertical line $Y O Y^{\prime}$ as the $x$-axis and $y$-axis, respectively. Graph of $2 x-3 y-17=0$ $\begin{array}{l} 2 x-3...
Solution: On a graph paper, draw a horizontal line $X^{\prime} O X$ and a vertical line YOY' as the $x$-axis and $y$-axis, respectively. Graph of $4 x-5 y+16=0$ $\begin{array}{l} 4 x-5 y+16=0 \\...
-axis.fSolution: Draw a horizontal line on a graph paper $\mathrm{X}^{\prime} \mathrm{OX}$ and a vertical line YOY' as the $\mathrm{x}$-axis and $\mathrm{y}$-axis, respectively. Graph of...
Solution: Draw a horizontal line on a graph paper X'OX and a vertical line YOY' as the $\mathrm{x}$-axis and $\mathrm{y}$-axis, respectively. Graph of $4 x-3 y+4=0$ $\begin{array}{l} 4 x-3 y+4=0 \\...
-axis.Solution: From the first eq., write y in terms of $x$ $\mathrm{y}=\frac{2 x+4}{3}\dots \dots(i)$ Substituting different values of $x$ in eq.(i) to get different values of $y$ For $x=-2,...
-axis.Solution: From the first eq., write y in terms of $\mathrm{x}$ $y=x+3\dots \dots(i)$ Substituting different values of $x$ in eq(i) to get different values of $y$ For $x=-3, y=-3+3=0$ For $x=-1,...
Solution: Draw a horizontal line on a graph paper X'OX and a vertical line YOY' as the $\mathrm{x}$-axis and $\mathrm{y}$-axis, respectively. $\begin{array}{l} x+2 y+2=0 \\ \Rightarrow 2 y=(-2-x) \\...
Solution: Draw a horizontal line oon a graph paper X'OX and a vertical line YOY' as the $x$-axis and $y$-axis, respectively. Graph of $2 x+3 y=4$ $2 x+3 y=4$ $\Rightarrow 3 y=(4-2 x)$ $\therefore...
Solution: From the first eq., write y in terms of $x$ $y=\frac{2 x+13}{3}\dots \dots (i)$ Substituting different values of $x$ in eq.(i) to get different values of $y$ For $x=-5,...
Solution: From the first eq., write y in terms of $x$ $y=-\left(\frac{5+2 x}{3}\right)\dots \dots (i)$ Substitute different values of $\mathrm{x}$ in eq.(i) to get different values of $\mathrm{y}$...
Solution: Draw a horizontal line on a graph paper X'OX and a vertical line YOY' as the $\mathrm{x}$-axis and y-axis, respectively. Graph of $3 \mathbf{x}+\mathbf{y}+\mathbf{1}=\mathbf{0}$...
Solution: The given eq. are: $3 x+2 y=12 \dots\dots (i)$ $5 x-2 y=4 \dots \dots(ii)$ From eq.(i), write y in terms of $x$ $\mathrm{y}=\frac{12-3 x}{2}\dots \dots(iii)$ Substitute different values of...
Solution: Draw a horizontal line on a graph paper X'OX and a vertical line YOY' as the $\mathrm{x}$-axis and y-axis, respectively. $\begin{array}{l} 2 x-5 y+4=0 \\ \Rightarrow 5 y=(2 x+4) \\...
Solution: Draw a horizontal line on a graph paper X'OX and a vertical line YOY' as the $\mathrm{x}$-axis and $\mathrm{y}$-axis, respectively. $\begin{array}{l} 2 x+3 y=8 \\ \Rightarrow 3 y=(8-2 x)...
Solution: Draw a horizontal line on a graph paper X'OX and a vertical line YOY' representing the xaxis and $y$-axis, respectively. Graph of $3 x+2 y=4$ $3 x+2 y=4$ $\Rightarrow 2 y=(4-3 x)$...
Solution: Draw a horizontal line on a graph paper$\mathrm{X}^{\prime} \mathrm{OX}$ and a vertical line YOY' representing the $\mathrm{x}-$ axis and $\mathrm{y}$-axis, respectively. $\begin{array}{l}...