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Function

Let f, g, h be real functions given by f(x) = sin x, g (x) =
$\mathbf{2x}$
and h (x) = cos x. Prove that fog = go (f h).

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

Given that f(x) = sin x, g (x) = \[\mathbf{2x}\] and h (x) = cos x We know that \[f:~R\to [-1,~1]~\]and g: R→ R Clearly, the range of g is a subset of the domain of f. fog: R → R Now, (f h) (x) =...

If f(x) = sin x and g(x) =
$\mathbf{2x}$
be two real functions, then describe gof and fog. Are these equal functions?

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

Given f(x) = sin x and g(x) = \[\mathbf{2x}\] We know that \[f:~R\to ~[-1,~1]~\]and g: R→ R Clearly, the range of f is a subset of the domain of g. gof: R→ R (gof) (x) = g (f (x)) = g (sin x)...

If f(x) =
$\mathbf{2x}~+\text{ }\mathbf{5}$
and g(x) =
${{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{1}$
be two real functions, then describe each of the following functions: (i) fog (ii) gof (iii) fof (iv)
${{\mathbf{f}}^{\mathbf{2}}}$
Also, show that
$\mathbf{fof}~\ne ~{{\mathbf{f}}^{\mathbf{2}}}$

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

f(x) and g(x) are polynomials. ⇒ f: R → R and g: R → R. So, fog: R → R and gof: R → R. (i) (fog) (x) = f (g (x)) = \[f\text{ }({{x}^{2}}~+\text{ }1)\] = \[2\text{ }({{x}^{2~}}+\text{ }1)\text{...

If f(x) =
$\left| x \right|$
, prove that fof = f

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

Let f(x) =
${{\mathbf{x}}^{\mathbf{2}}}~+~\mathbf{x}~+\text{ }\mathbf{1}$
and g(x) = sin x. Show that fog ≠ gof.

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

Given f(x) = \[{{\mathbf{x}}^{\mathbf{2}}}~+~\mathbf{x}~+\text{ }\mathbf{1}\]and g(x) = sin x Now we have to prove fog ≠ gof (fog) (x) = f (g (x)) = f (sin x) = \[si{{n}^{2~}}x~+~sin~x~+~1\]...

Find fog and gof, if f(x) =
${{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2}$
, g (x) =
$\mathbf{1}\text{ }-\text{ }\mathbf{1}/\text{ }\left( \mathbf{1}-\mathbf{x} \right)$

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

Given f (x) = \[{{x}^{2}}+~2\]and \[g~\left( x \right)~=~1\text{ }\text{ }1\text{ }/\text{ }\left( 1\text{ }\text{ }x \right)\] \[f:~R~\to ~[~2,~\infty ~)\] For domain of \[g:~1-~x~\ne ~0\]...

Find fog and gof, if
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{c},\text{ }\mathbf{c}\in \mathbf{R}$
, g(x) =
$\mathbf{sin}\text{ }{{\mathbf{x}}^{\mathbf{2}}}$

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

Given f (x) = c, g (x) = \[sin~{{x}^{2}}\] f: R → {c} ; \[g:~R\to ~[~0,~1~]\] Now we have to compute fog Clearly, the range of g is a subset of the domain of f. fog: R→R (fog) (x) = f (g (x))...

Find fog and gof, if f(x)=
$x+1$
, g (x) =
$\mathbf{2x}~+~\mathbf{3}$

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

Given f (x) = \[x+1\], g (x) = \[2x~+~3\] f: R→R ; g: R → R Now we have to compute fog Clearly, the range of g is a subset of the domain of f. ⇒ fog: R→ R (fog) (x) = f (g (x)) = \[f~\left( 2x+3...

Find fog and gof, if f (x) =
$x+1$
, g (x) = sin x

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

Given \[f\left( x \right)~=~x+1\], g(x) = sin x f: R→R ; \[g:~R\to [-1,~1]\] Now we have to compute fog Clearly, the range of g is a subset of the domain of f. Set of the domain of f. ⇒ fog: R→ R...

Find fog and gof, if f (x) =
${{\sin }^{-1}}x$
, g(x) =
${{x}^{2}}$

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

Given f (x) = \[sin{{~}^{-1}}~x\], g(x) = \[{{x}^{2}}\] \[f:~\left[ -1,1 \right]\to [(-\pi )/2\text{ },\pi /2];~g~:~R~\to ~[0,~\infty )\] Now we have to compute fog:...

Find fog and gof, if f (x) =
$x+1$
, g(x) =
${{\mathbf{e}}^{\mathbf{x}}}$

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

Given f (x) = \[x\text{ }+\text{ }1\], g(x) = \[{{e}^{x}}\] \[f:~R\to R~;~g:~R~\to ~[~1,~\infty )\] Now we have calculate fog: Clearly, range of g is a subset of domain of f. ⇒ fog: R→R...

Find fog and gof, if f (x) =
$\left| x \right|$
, g (x) = sin x

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

Given \[f~\left( x \right)~=~\left| x \right|\],g(x) = sin x \[f:~R~\to ~(0,~\infty )~;~g~:~R\to [-1,~1]\] Now we have to calculate fog, Clearly, the range of g is a subset of the domain of f....

Find fog and gof, if f (x) =
${{x}^{2}}$
, g (x) = cos x

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

f (x) = \[{{x}^{2}}\], g(x) = cos x \[f:~R\to ~[0,~\infty )~\]; \[g:~R\to [-1,~1]\] Now we have to calculate fog, Clearly, the range of g is not a subset of the domain of f. ⇒ Domain (fog) = {x: x ∈...

Find fog and gof, if f (x) =
${{\mathbf{e}}^{\mathbf{x}}}$
, g (x) =
$\mathbf{lo}{{\mathbf{g}}_{\mathbf{e}}}~\mathbf{x}$

CBSE, Class 12, Exercise 2.3, Function, Maths, RD Sharma

Given f (x) = \[{{\mathbf{e}}^{\mathbf{x}}}\], g(x) = \[\mathbf{lo}{{\mathbf{g}}_{\mathbf{e}}}~\mathbf{x}\] Let \[f:~R~\to ~(0,~\infty )\];and \[g:~(0,~\infty )~\to ~R\] Now we have to calculate...

Let f: R → R and g: R → R be defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{2}}}~$
and
$\mathbf{g}\left( \mathbf{x} \right)\text{ }=~\mathbf{x}~+\text{ }\mathbf{1}$
. Show that
$\mathbf{fog}~\ne ~\mathbf{gof}$
.

CBSE, Class 12, Exercise 2.2, Function, Maths, RD Sharma

Given f: R → R and g: R → R. So, the domains of f and g are the same. Consider (fog) (x) = f (g (x)) = \[f~\left( x\text{ }+\text{ }1 \right)\text{ }=~{{\left( x\text{ }+\text{ }1 \right)}^{2}}\]...

Let
${{\mathbf{R}}^{+}}$
.be the set of all non-negative real numbers. If
$\mathbf{f}:~{{\mathbf{R}}^{+}}~\to ~{{\mathbf{R}}^{+}}~$
and
$\mathbf{g}~:~{{\mathbf{R}}^{+}}~\to ~{{\mathbf{R}}^{+}}~$
are defined as
$\mathbf{f}\left( \mathbf{x} \right)={{\mathbf{x}}^{\mathbf{2}}}~$
and
$\mathbf{g}\left( \mathbf{x} \right)=+\text{ }\sqrt{\mathbf{x}}$
, find fog and gof. Are they equal functions.

CBSE, Class 12, Exercise 2.2, Function, Maths, RD Sharma

Given \[\mathbf{f}:~{{\mathbf{R}}^{+}}~\to ~{{\mathbf{R}}^{+}}\]and \[\mathbf{g}~:~{{\mathbf{R}}^{+}}~\to ~{{\mathbf{R}}^{+}}~\] So, \[fog:~{{R}^{+}}~\to ~{{R}^{+}}~\]and \[gof:~{{R}^{+}}~\to...

Find
$\mathbf{fog}~\left( \mathbf{2} \right)$
and
$\mathbf{gof}~\left( \mathbf{1} \right)$
when f: R → R;
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{8}$
and g: R → R;
$\mathbf{g}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{1}$
.

CBSE, Class 12, Exercise 2.2, Function, Maths, RD Sharma

Given f: R → R; \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{8}\]and g: R → R; \[\mathbf{g}\left( \mathbf{x} \right)\text{ }=\text{...

Let A = {a, b, c}, B = {u, v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as: f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}. Show that f and g both are bijections and find fog and gof.

CBSE, Class 12, Exercise 2.2, Function, Maths, RD Sharma

Given f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}. Also given that A = {a, b, c}, B = {u, v, w} Now we have to show f and g both are bijective. Consider f = {(a, v), (b, u), (c, w)}...

Let
$\mathbf{f}~=\text{ }\left\{ \left( \mathbf{1},\text{ }-\mathbf{1} \right),\text{ }\left( \mathbf{4},\text{ }-\mathbf{2} \right),\text{ }\left( \mathbf{9},\text{ }-\mathbf{3} \right),\text{ }\left( \mathbf{16},\text{ }\mathbf{4} \right) \right\}$
and
$\mathbf{g}~=\text{ }\left\{ \left( -\mathbf{1},\text{ }-\mathbf{2} \right),\text{ }\left( -\mathbf{2},\text{ }-\mathbf{4} \right),\text{ }\left( -\mathbf{3},\text{ }-\mathbf{6} \right),\text{ }\left( \mathbf{4},\text{ }\mathbf{8} \right) \right\}$
.Show that gof is defined while fog is not defined. Also, find gof.

CBSE, Class 12, Exercise 2.2, Function, Maths, RD Sharma

Given \[\mathbf{f}~=\text{ }\left\{ \left( \mathbf{1},\text{ }-\mathbf{1} \right),\text{ }\left( \mathbf{4},\text{ }-\mathbf{2} \right),\text{ }\left( \mathbf{9},\text{ }-\mathbf{3} \right),\text{...

Let
$\mathbf{f}~=\text{ }\left\{ \left( \mathbf{3},\text{ }\mathbf{1} \right),\text{ }\left( \mathbf{9},\text{ }\mathbf{3} \right),\text{ }\left( \mathbf{12},\text{ }\mathbf{4} \right) \right\}$
and
$\mathbf{g}~=\text{ }\left\{ \left( \mathbf{1},\text{ }\mathbf{3} \right),\text{ }\left( \mathbf{3},\text{ }\mathbf{3} \right)\text{ }\left( \mathbf{4},\text{ }\mathbf{9} \right)\text{ }\left( \mathbf{5},\text{ }\mathbf{9} \right) \right\}$
. Show that gof and fog are both defined. Also, find fog and gof.

CBSE, Class 12, Exercise 2.2, Function, Maths, RD Sharma

Given \[\mathbf{f}~=\text{ }\left\{ \left( \mathbf{3},\text{ }\mathbf{1} \right),\text{ }\left( \mathbf{9},\text{ }\mathbf{3} \right),\text{ }\left( \mathbf{12},\text{ }\mathbf{4} \right)...

Find gof and fog when f: R → R and g : R → R is defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{8}{{\mathbf{x}}^{\mathbf{3}}}~$
and
$\mathbf{g}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{1}/\mathbf{3}}}$

CBSE, Class 12, Exercise 2.2, Function, Maths, RD Sharma

Given, f: R → R and g: R → R So, gof: R → R and fog: R → R \[f\left( x \right)\text{ }=\text{ }8{{x}^{3}}~\]and \[g\left( x \right)\text{ }=~{{x}^{1/3}}\] (gof) (x) = g (f (x)) = \[g~(8{{x}^{3}})\]...

Find gof and fog when f: R → R and g : R → R is defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2x}~-\text{ }\mathbf{3}$
and
$\mathbf{g}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{3x}~-\text{ }\mathbf{4}~$

CBSE, Class 12, Exercise 2.2, Function, Maths, RD Sharma

Given, f: R → R and g: R → R So, gof: R → R and fog: R → R \[f\left( x \right)\text{ }=~{{x}^{2}}~+\text{ }2x~-\text{ }3\] and \[g\left( x \right)\text{ }=\text{ }3x~-\text{ }4\] (gof) (x)...

Find gof and fog when f: R → R and g : R → R is defined by f (x) = x and
$\mathbf{g}\left( \mathbf{x} \right)\text{ }=\text{ }\left| \mathbf{x} \right|~$

CBSE, Class 12, Exercise 2.2, Function, Maths, RD Sharma

Given, f: R → R and g: R → R So, gof: R → R and fog: R → R f(x) = x and \[g\left( x \right)\text{ }=\text{ }\left| x \right|\] (gof) (x) = g (f (x)) = g (x) = \[\left| x \right|\] Now (fog) (x)...

Find gof and fog when f: R → R and g : R → R is defined by
$\mathbf{f}~\left( \mathbf{x} \right)~=~{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{8}~$
and
$\mathbf{g}\left( \mathbf{x} \right)~=\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{1}$

CBSE, Class 12, Exercise 2.2, Function, Maths, RD Sharma

Given, f: R → R and g: R → R So, gof: R → R and fog: R → R \[f\left( x \right)\text{ }=~{{x}^{2}}~+\text{ }8~\]and \[g\left( x \right)\text{ }=\text{ }3{{x}^{3}}~+\text{ }1\] (gof) (x) = g (f (x))...

Find gof and fog when f: R → R and g : R → R is defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{2x}~+~{{\mathbf{x}}^{\mathbf{2}}}~$
and
$\mathbf{g}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{3}}}$

CBSE, Class 12, Exercise 2.2, Function, Maths, RD Sharma

Given, f: R → R and g: R → R so, gof: R → R and fog: R → R \[f\left( x \right)\text{ }=\text{ }2x~+~{{x}^{2}}~\] and \[g\left( x \right)\text{ }=~{{x}^{3}}\] (gof) (x)= g (f (x))...

Find gof and fog when f: R → R and g : R → R is defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{2x}~+\text{ }\mathbf{3}$
and
$\mathbf{g}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{5}$
.

CBSE, Class 12, Exercise 2.2, Function, Maths, RD Sharma

Given, f: R → R and g: R → R So, gof: R → R and fog: R → R Also given that \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{2x}~+\text{ }\mathbf{3}\]and \[\mathbf{g}\left( \mathbf{x}...

If f: R → R be the function defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{4}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{7}$
, show that f is a bijection.

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Let
$A~=\text{ }\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3} \right\}$
. Write all one-one from A to itself.

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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 =...

Are the following set of ordered pair of a function? If so, examine whether the mapping is injective or surjective: (i) {(x, y): x is a person, y is the mother of x} (ii) {(a, b): a is a person, b is an ancestor of a}

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Let
$A~=\text{ }\left[ -\mathbf{1},\text{ }\mathbf{1} \right]$
. Then, discuss whether the following function from A to itself is one-one, onto or bijective: (i) f (x) = x/
$2$
(ii)
$\mathbf{g}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }\left| \mathbf{x} \right|$
(iii)
$\mathbf{h}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }{{\mathbf{x}}^{\mathbf{2}}}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

(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)...

Show that the function f: R − {
$3$
} → R − {
$2$
} given by f(x) =
$\left( \mathbf{x}-\mathbf{2} \right)/\left( \mathbf{x}-\mathbf{3} \right)~$
is a bijection.

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

If f: A → B is an injection, such that range of f = {a}, determine the number of elements in A.

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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,...

Classify the following function as injection, surjection or bijection: f: R → R, defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{x}/({{\mathbf{x}}^{\mathbf{2}~}}+\text{ }\mathbf{1})$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

) 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...

Classify the following function as injection, surjection or bijection: f: R → R, defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{1}\text{ }+~{{\mathbf{x}}^{\mathbf{2}}}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Classify the following function as injection, surjection or bijection: f: R → R, defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{5}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{4}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Classify the following function as injection, surjection or bijection: f: R → R, defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{5}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{4}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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:...

Classify the following function as injection, surjection or bijection: f: Q → Q, defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{1}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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:...

Classify the following function as injection, surjection or bijection:
$\mathbf{f}:~\mathbf{Q}~-\text{ }\left\{ \mathbf{3} \right\}\text{ }\to ~\mathbf{Q}$
, defined by
$\mathbf{f}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }\left( \mathbf{2x}\text{ }+\mathbf{3} \right)/\left( \mathbf{x}-\mathbf{3} \right)$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Classify the following function as injection, surjection or bijection: f: R → R, defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}\mathbf{x}~+\text{ }\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}\mathbf{x}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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....

Classify the following function as injection, surjection or bijection: f: R → R, defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{3}}}~-~\mathbf{x}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Classify the following function as injection, surjection or bijection: f: R → R, defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{1}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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:...

Classify the following function as injection, surjection or bijection: f: R → R, defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{sin}\text{ }\mathbf{x}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Classify the following function as injection, surjection or bijection: f: Z → Z, defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=~\mathbf{x}~-\text{ }\mathbf{5}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Classify the following function as injection, surjection or bijection: f: Z → Z, defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{2}}}~+~\mathbf{x}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Classify the following function as injection, surjection or bijection: f: R → R, defined by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\left| \mathbf{x} \right|$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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:...

Classify the following function as injection, surjection or bijection: f: Z → Z given by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{3}}}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Classify the following function as injection, surjection or bijection: f: N → N given by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{3}}}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Classify the following function as injection, surjection or bijection: f: Z → Z given by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{2}}}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Classify the following function as injection, surjection or bijection: f: N → N given by
$\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{2}}}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Let
$\mathbf{A}~=\text{ }\left\{ -\mathbf{1},\text{ }\mathbf{0},\text{ }\mathbf{1} \right\}$
and
$\mathbf{f}~=\text{ }\{(\mathbf{x},~{{\mathbf{x}}^{\mathbf{2}}})\text{ }:~\mathbf{x}~\in ~\mathbf{A}\}$
. Show that f : A → A is neither one-one nor onto.

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Prove that the function f: N → N, defined by f(x) =
${{\mathbf{x}}^{\mathbf{2}}}~+~\mathbf{x}~+\text{ }\mathbf{1}$
, is one-one but not onto

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Which of the following functions from A to B are one-one and onto?
${{\mathbf{f}}_{\mathbf{3}}}~=\text{ }\left\{ \left( \mathbf{a},~\mathbf{x} \right),\text{ }\left( \mathbf{b},~\mathbf{x} \right),\text{ }\left( \mathbf{c},~\mathbf{z} \right),\text{ }\left( \mathbf{d},~\mathbf{z} \right) \right\};~\mathbf{A}~=\text{ }\left\{ \mathbf{a},~\mathbf{b},~\mathbf{c},~\mathbf{d}, \right\},~\mathbf{B}~=\text{ }\left\{ \mathbf{x},~\mathbf{y},~\mathbf{z} \right\}$
.

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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...

Which of the following functions from A to B are one-one and onto?
${{\mathbf{f}}_{\mathbf{2}}}~=\text{ }\left\{ \left( \mathbf{2},~\mathbf{a} \right),\text{ }\left( \mathbf{3},~\mathbf{b} \right),\text{ }\left( \mathbf{4},~\mathbf{c} \right) \right\};~\mathbf{A}~=\text{ }\left\{ \mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4} \right\},~\mathbf{B}~=\text{ }\left\{ \mathbf{a},~\mathbf{b},~\mathbf{c} \right\}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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)...

Which of the following functions from A to B are one-one and onto?
${{\mathbf{f}}_{\mathbf{1}}}~=\text{ }\left\{ \left( \mathbf{1},\text{ }\mathbf{3} \right),\text{ }\left( \mathbf{2},\text{ }\mathbf{5} \right),\text{ }\left( \mathbf{3},\text{ }\mathbf{7} \right) \right\};~\mathbf{A}~=\text{ }\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3} \right\},~\mathbf{B}~=\text{ }\left\{ \mathbf{3},\text{ }\mathbf{5},\text{ }\mathbf{7} \right\}$

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

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}...

Give an example of a function (i) Which is one-one but not onto. (ii) Which is not one-one but onto. (iii) Which is neither one-one nor onto.

CBSE, Class 12, Exercise 2.1, Function, Maths, RD Sharma

(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)....

A function $f: R \rightarrow R$ is defined as $f(x)=x^{3}+4 .$ Is it a bijection or not? In case it is a bijection, find $\mathbf{f}^{-1}$ (3).

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given that $f: R \rightarrow R$ is defined as $f(x)=x^{3}+4$ Injectivity of f: Let $x$ and $y$ be two elements of domain (R), Such that $f(x)=f(y)$ $\Rightarrow x^{3}+4=y^{3}+4$ $\Rightarrow...

If $f: R \rightarrow R$ be defined by $f(x)=x^{3}-3$ , then prove that $f^{-1}$ exists and find a formula for $f^{-1}$ . Hence, find $f^{-1}(24)$ and $f^{-1}(5)$ .

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given $f: R \rightarrow R$ be defined by $f(x)=x^{3}-3$ Now we have to prove that $\mathrm{f}^{-1}$ exists Injectivity of f: Let $x$ and $y$ be two elements in domain $(R)$, Such that,...

Consider f: $R^{+} \rightarrow[-5, \infty)$ given by $f(x)=9 x^{2}+6 x-5 .$ Show that $f$ is invertible with $f^{-1}(x)=(v(x+6)-1) / 3$

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given $f: R^{+} \rightarrow[-5, \infty)$ given by $f(x)=9 x^{2}+6 x-5$ We have to show that $\mathrm{f}$ is invertible. \section{Injectivity of f:} Let $x$ and $y$ be two elements of domain...

If $f(x)=(4 x+3) /(6 x-4), x \neq(2 / 3)$ show that $f o f(x)=x$ , for all $x \neq(2 / 3)$ . What is the inverse of f?

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

It is given that $f(x)=(4 x+3) /(6 x-4), x \neq 2 / 3$ Now we have to show $\operatorname{fof}(x)=x$ $($ fof $)(x)=f(f(x))$ $=f((4 x+3) /(6 x-4))$ $=(4((4 x+3) /(6 x-4))+3) /(6((4 x+3) /(6 x-4))-4)$...

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

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given $f: R \rightarrow R^{+} \rightarrow[4, \infty)$ given by $f(x)=x^{2}+4$ Now we have to show that $f$ is invertible, Consider injection of f: \section{RD Sharma Solutions for Class 12 Maths...

Consider $f: R \rightarrow R$ given by $f(x)=4 x+3 .$ Show that $f$ is invertible. Find the inverse of $\mathbf{f}$ .

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given $f: R \rightarrow R$ given by $f(x)=4 x+3$ Now we have to show that the given function is invertible. Consider injection of f: Let $x$ and $y$ be two elements of domain $(R)$, Such that...

Show that the function $f: Q \rightarrow Q$ , defined by $f(x)=3 x+5$ , is invertible. Also, find $f^{-1}$

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given function $f: Q \rightarrow Q$, defined by $f(x)=3 x+5$ Now we have to show that the given function is invertible. Injection of f: Let $x$ and $y$ be two elements of the domain (Q), Such that...

Let $A=\{1,2,3,4\} ; B=\{3,5,7,9\} ; C=\{7,23,47,79\}$ and f: $A \rightarrow B, g: B \rightarrow C$ be defined as $f(x)=2 x+1$ and $g(x)=x^{2}-2 .$ Express $(\text { gof })^{-1}$ and $f^{-1} o g^{-1}$ as the sets of ordered pairs and verify that (gof) $^{-1}=\mathbf{f}^{-1} \mathrm{og}^{-1}$ .

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

$\Rightarrow \mathrm{f}=\{(1,2(1)+1),(2,2(2)+1),(3,2(3)+1),(4,2(4)+1)\}$ $=\{(1,3),(2,5),(3,7),(4,9)\}$ Also given that $g(x)=x^{2}-2$ $\Rightarrow...

Consider $f:\{1,2,3\} \rightarrow\{a, b, c\}$ and $g:\{a, b, c\} \rightarrow\{$ 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 $\mathbf{f}^{-1}, \mathrm{~g}^{-1}$ and gof $^{-1}$ and show that $(\text { gof })^{-1}=f^{-1} 0 \mathrm{~g}^{-1}$

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

Given $f=\{(1, a),(2, b),(c, 3)\}$ and $g=\{(a$, apple), $(b, b a l l),(c$, cat $)\}$ Clearly, $f$ and $g$ are bijections. So, $f$ and $g$ are invertible. Now, $f^{-1}=\{(a, 1),(b, 2),(3, c)\}$ and...

Find $f^{-1}$ if it exists: f: A $\rightarrow$ , where (i) $A=\{0,-1,-3,2\} ; B=\{-9,-3,0,6\}$ and $f(x)=3 x$ (ii) $A=\{1,3,5,7,9\} ; B=\{0,1,9,25,49,81\}$ and $f(x)=x^{2}$

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

(i) Given $A=\{0,-1,-3,2\} ; B=\{-9,-3,0,6\}$ and $f(x)=3 x$. So, $f=\{(0,0),(-1,-3),(-3,-9),(2,6)\}$ \section{RD Sharma Solutions for Class 12 Maths Chapter 2 Function} Here, different elements of...

State with reason whether the following functions have inverse: (iii) $h:\{2,3,4,5\} \rightarrow\{7,9,11,13\}$ with $h=\{(2,7),(3,9),(4,11),(5,13)\}$

CBSE, Exercise 2.4, Function, Maths, RD Sharma

iii) Given $\mathrm{h}:\{2,3,4,5\} \rightarrow\{7,9,11,13\}$ with $\mathrm{h}=\{(2,7),(3,9),(4,11),(5,13)\}$ different elements of the domain have different images in the co-domain. $\Rightarrow...

State with reason whether the following functions have inverse: (i) $f:\{1,2,3,4\} \rightarrow\{10\}$ with $f=\{(1,10),(2,10),(3,10),(4,10)\}$ (ii) $g:\{5,6,7,8\} \rightarrow\{1,2,3,4\}$ with $g=\{(5,4),(6,3),(7,4),(8,2)\}$

CBSE, Class 12, Exercise 2.4, Function, Maths, RD Sharma

(i) Given $\mathrm{f}:\{1,2,3,4\} \rightarrow\{10\}$ with $f=\{(1,10),(2,10),(3,10),(4,10)\}$ We have: $f(1)=f(2)=f(3)=f(4)=10$ $\Rightarrow \mathrm{f}$ is not one-one. $\Rightarrow \mathrm{f}$ is...