Exercise 2.2 Archives - Noon Academy

Exercise 2.2

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}...

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
$\mathbf{fog}~\ne ~\mathbf{gof}$
.

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.

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
$\mathbf{g}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{1}/\mathbf{3}}}$

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
$\mathbf{g}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{3x}~-\text{ }\mathbf{4}~$

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

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
$\mathbf{g}\left( \mathbf{x} \right)~=\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{1}$

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
$\mathbf{g}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{3}}}$

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
$\mathbf{g}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{5}$
.

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