(i) from the question, {(x, y): x is a person, y is the mother of x} So we can say that each person ‘x’ has only one biological mother. Therefore, the above set of ordered pairs make a function. If...
![\[\mathbf{g}\text{ }=\text{ }\left\{ \left( \mathbf{1},\text{ }\mathbf{1} \right),\text{ }\left( \mathbf{2},\text{ }\mathbf{3} \right),\text{ }\left( \mathbf{3},\text{ }\mathbf{5} \right),\text{ }\left( \mathbf{4},\text{ }\mathbf{7} \right) \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d072d1f94a4403000866fa1969963f5e_l3.png "Rendered by QuickLaTeX.com")
![\[\mathbf{g}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{\alpha x}\text{ }+\text{ }\mathbf{\beta }\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e132af380935f39045049cfed6bc479e_l3.png "Rendered by QuickLaTeX.com")
![\[\mathbf{\alpha }\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2290b3b6ebb8da39f941931b782a55ee_l3.png "Rendered by QuickLaTeX.com")
![\[\mathbf{\beta }\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-24941173bdd7b0c67a5c814c4779615f_l3.png "Rendered by QuickLaTeX.com")
From the question, \[\mathbf{g}\text{ }=\text{ }\left\{ \left( \mathbf{1},\text{ }\mathbf{1} \right),\text{ }\left( \mathbf{2},\text{ }\mathbf{3} \right),\text{ }\left( \mathbf{3},\text{ }\mathbf{5}...
![\[\mathbf{f}:\text{ }\mathbf{R}\text{ }\to \text{ }\mathbf{R}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-dd6ed4993146108e6223c8db2541e3f7_l3.png "Rendered by QuickLaTeX.com")
![\[\mathbf{f}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{3x}\text{ }+\text{ }\mathbf{2}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-995851d8d7836e7b40ef98cf0bd42f1a_l3.png "Rendered by QuickLaTeX.com")
Given, \[\mathbf{f}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{3x}\text{ }+\text{ }\mathbf{2}\] Then, \[\begin{array}{*{35}{l}} f\text{ }\left(...
![\[\mathbf{f}{{~}^{\mathbf{1}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-767f1094114853260386f6d809284ad0_l3.png "Rendered by QuickLaTeX.com")
According to the question, A = {a, b, c, d} and f = {(a, b), (b, d), (c, a), (d, c)} So, \[\mathbf{f}{{~}^{\mathbf{1}}}\] = {(b, a), (d, b), (a, c), (c, d)}
![\[\mathbf{2x}\text{ }\text{ }\mathbf{3}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-022296614ad7968c0c3b37dedea5273e_l3.png "Rendered by QuickLaTeX.com")
![\[\forall \mathbf{x}\in \mathbf{R}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-93681e8b3dafd096e20967b3836271fa_l3.png "Rendered by QuickLaTeX.com")
According to the question the function, f (x) = \[\mathbf{2x}\text{ }\text{ }\mathbf{3}\], \[\forall \mathbf{x}\in \mathbf{R}\] Let us consider \[y\text{ }=\text{ }2x\text{ }\text{ }3\] \[x\text{...
![\[\mathbf{g}:\text{ }\mathbf{R}\text{ }\to \text{ }\mathbf{R}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2a6e4f932765ad5e387a51e3c62eeb38_l3.png "Rendered by QuickLaTeX.com")
![\[\mathbf{2x}\text{ }+\text{ }\mathbf{1}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-16fd72fb9940c8b247bd1589bb6b4d2a_l3.png "Rendered by QuickLaTeX.com")
![\[{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{2}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e52890f89dba1d3f978c6f41597f9bd7_l3.png "Rendered by QuickLaTeX.com")
![\[\mathbf{g}\text{ }\mathbf{o}\text{ }\mathbf{f}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-446f4667cf4569e1d8763649734e7995_l3.png "Rendered by QuickLaTeX.com")
According to the question, f(x) = \[\mathbf{2x}\text{ }+\text{ }\mathbf{1}\]and g (x) = \[{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{2}\], \[\forall \mathbf{x}\in \mathbf{R}\] Thus, \[g\text{...
![\[\sqrt{(\mathbf{25}\text{ }\text{ }{{\mathbf{x}}^{\mathbf{2}}})}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-653a123eb2d1b6788fe0e3cb5438b845_l3.png "Rendered by QuickLaTeX.com")
According to the question, f(x) = \[\sqrt{(\mathbf{25}\text{ }\text{ }{{\mathbf{x}}^{\mathbf{2}}})}\] The function is defined if \[25\text{ }\text{ }{{x}^{2}}~\ge \text{ }0\] So, \[{{x}^{2}}~\le...
According to the given question the relation, R = {(a, a), (b, c), (a, b)} Therefore, to make R as reflexive we should add (b, b) and (c, c) to R. Also, to make R as transitive we should add (a, c)...
, show that 
Given function is: $y=\left(\tan ^{-1} x\right)^{2}$.....(i) Representing $y_{2}$ as second derivative of the function and $y_{1}$ as first derivative and , we get, $y_{1}=2\left(\tan ^{-1} x\right)...
and 
, find 
.Solution: The provided expressions are $x=a(\cos t+t \sin t)$and $y=a(\sin t-t \cos t)$ $x=a(\cos t+t \sin t)$ With respect to $t$ differentiating both the sides $\frac{d x}{d t}=a\left(-\sin...
Let consider the centre of the circle is O. Given in the question- OP = radius = $5cm$ OQ = $13cm$ At point P, a tangent if formed such that the line passing through O intersects it at point Q....