Given f: R → R is a function defined by \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{4}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{7}\] Injectivity: Let x and y be any two elements...

Given \[A~=\text{ }\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3} \right\}\] Number of elements in  A = \[3\] Number of one-one functions = number of ways of arranging \[3\] elements =...

Let f = {(x, y): x is a person, y is the mother of x} As, for each element x in domain set, there is a unique related element y in co-domain set. So, f is the function. Injection test: As, y can be...

(i) Given f: A → A, given by f (x) = x/ \[2\] Now we have to show that the given function is one-one and on-to Injection test: Let x and y be any two elements in the domain (A), such that f(x)...

Given that \[f:~R~-\text{ }\left\{ 3 \right\}\text{ }\to ~R~-\text{ }\left\{ 2 \right\}\]given by f (x) = \[\left( \mathbf{x}-\mathbf{2} \right)/\left( \mathbf{x}-\mathbf{3} \right)~\] Now we have...

Given f: A → B is an injection And also given that range of f = {a} So, the number of images of  f = \[1\] Since, f  is an injection, there will be exactly one image for each element of f . So,...

) Given f: R → R, defined by \[f\left( x \right)\text{ }=\text{ }x/({{x}^{2~}}+\text{ }1)\] Now we have to check for the given function is injection, surjection and bijection condition. Injection...

Given f: R → R, defined by \[f\left( x \right)\text{ }=\text{ }1\text{ }+~{{x}^{2}}\] Now we have to check for the given function is injection, surjection and bijection condition. Injection...

Given f: R → R, defined by \[f\left( x \right)\text{ }=\text{ }5{{x}^{3}}~+\text{ }4\] Now we have to check for the given function is injection, surjection and bijection condition. Injection...

Given f: R → R, defined by \[f\left( x \right)\text{ }=\text{ }5{{x}^{3}}~+\text{ }4\] Now we have to check for the given function is injection, surjection and bijection condition. Injection test:...

Given f: Q → Q, defined by \[f\left( x \right)\text{ }=~{{x}^{3}}~+\text{ }1\] Now we have to check for the given function is injection, surjection and bijection condition. Injection test:...

Given \[f:~Q~-\text{ }\left\{ 3 \right\}\text{ }\to ~Q\], defined by \[f\text{ }\left( x \right)\text{ }=\text{ }\left( 2x\text{ }+3 \right)/\left( x-3 \right)\] Now we have to check for the given...

Given f: R → R, defined by \[f\left( x \right)\text{ }=\text{ }si{{n}^{2}}x~+\text{ }co{{s}^{2}}x\] Now we have to check for the given function is injection, surjection and bijection condition....

Given f: R → R, defined by \[f\left( x \right)\text{ }=~{{x}^{3}}~-~x\] Now we have to check for the given function is injection, surjection and bijection condition. Injection test: Let x and y be...

Given f: R → R, defined by \[f\left( x \right)\text{ }=~{{x}^{3}}~+\text{ }1\] Now we have to check for the given function is injection, surjection and bijection condition. Injection test:...

Given f: R → R, defined by f(x) = sin x Now we have to check for the given function is injection, surjection and bijection condition. Injection test: Let x and y be any two elements in the domain...

Given f: Z → Z, defined by \[f\left( x \right)\text{ }=~x~\text{ }5\] Now we have to check for the given function is injection, surjection and bijection condition. Injection test: Let x and y be any...

Given f: Z → Z, defined by \[f\left( x \right)\text{ }=~{{x}^{2}}~+~x\] Now we have to check for the given function is injection, surjection and bijection condition. Injection test: Let x and y be...

Given f: R → R, defined by \[f\left( x \right)\text{ }=\text{ }\left| x \right|\] Now we have to check for the given function is injection, surjection and bijection condition. Injection test:...

Given f: Z → Z given by \[f\left( x \right)\text{ }=~{{x}^{3}}\] Now we have to check for the given function is injection, surjection and bijection condition. Injection condition: Let x and y be any...

Given f: N → N given by \[f\left( x \right)\text{ }=~{{x}^{3}}\] Now we have to check for the given function is injection, surjection and bijection condition. Injection condition: Let x and y be any...

Given f: Z → Z, given by \[~f\left( x \right)\text{ }=~{{x}^{2}}\] Now we have to check for the given function is injection, surjection and bijection condition. Injection condition: Let x and y be...

Given f: N → N, given by \[~f\left( x \right)\text{ }=~{{x}^{2}}\] Now we have to check for the given function is injection, surjection and bijection condition. Injection condition: Let x and y be...

Given \[\mathbf{A}~=\text{ }\left\{ -\mathbf{1},\text{ }\mathbf{0},\text{ }\mathbf{1} \right\}\]and \[\mathbf{f}~=\text{ }\{(\mathbf{x},~{{\mathbf{x}}^{\mathbf{2}}})\text{ }:~\mathbf{x}~\in...

Given f: N → N, defined by f(x) = \[{{\mathbf{x}}^{\mathbf{2}}}~+~\mathbf{x}~+\text{ }\mathbf{1}\] Now we have to prove that given function is one-one Injectivity: Let x and y be any two elements in...

Consider Injectivity: \[{{f}_{3}}~\left( a \right)\text{ }=\text{ }x\] \[{{f}_{3}}~\left( b \right)\text{ }=\text{ }x\] \[{{f}_{3}}~\left( c \right)\text{ }=\text{ }z\] \[{{f}_{3}}~\left( d...

Consider \[{{\mathbf{f}}_{\mathbf{2}}}~=\text{ }\left\{ \left( \mathbf{2},~\mathbf{a} \right),\text{ }\left( \mathbf{3},~\mathbf{b} \right),\text{ }\left( \mathbf{4},~\mathbf{c} \right)...

Consider \[{{\mathbf{f}}_{\mathbf{1}}}~=\text{ }\left\{ \left( \mathbf{1},\text{ }\mathbf{3} \right),\text{ }\left( \mathbf{2},\text{ }\mathbf{5} \right),\text{ }\left( \mathbf{3},\text{ }\mathbf{7}...

(i)Let f: Z → Z given by f(x) = \[3x~+\text{ }2\] Let us check one-one condition on f(x) = \[3x~+\text{ }2\] Injectivity: Let x and y be any two elements in the domain (Z), such that f(x) = f(y)....