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Graphing inverse functions Quiz 2

mathematics

 There are equations $x(t)$ and $y(t)$. Let the inverse of $x(t)$ be $p(x)$ and $y(t)$ be $q(y)$. Replace $x$ in $p(x)$ with $n$ i.e., eqn. changes to $p(n)$ and similarly replace $y$ in $q(y)$ with $n$, i.e., eqn. changes to $q(n)$. $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/fdaae723-f952-4cc0-89b6-e1b0a7a2de374821973856949225777.png"/> $ \\ $ Find the value of ${{\sum\limits_{n=0}^{n=5}{\left[ p\left( n \right) \right]}}^{2}}\left[ q\left( n \right) \right]$

 Use Venn diagram representation of function to find the inverse of the function. After finding the inverse, put the appropriate values in the equation.

$x\left[ t \right]$:  $ \\ $  <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/9663b54d-3f40-4647-be3b-aa85e0f2d36a6279788568018996273.png"/>   $ \\ $ $p\left[ n \right]$:  $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/8cf35bf4-dcbc-4d5b-a177-cf31e484e975252113448664633180.png"/>  $ \\ $ $y\left[ t \right]$:  $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/e0983f25-ecee-43ce-b1f2-d86dc4e4720f3999224250102909027.png"/> $ \\ $ $q\left[ n \right]$: $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/6588aeed-045a-471f-a1db-7182351c12ca805573098787217678.png"/>  $ \\  {{\sum\limits_{n=0}^{n=5}{\left[ p\left( n \right) \right]}}^{2}}\left[ q\left( n \right) \right] \\  ={{\left[ p\left( 0 \right) \right]}^{2}}\left[ q\left( 0 \right) \right]+{{\left[ p\left( 1 \right) \right]}^{2}}\left[ q\left( 1 \right) \right]+{{\left[ p\left( 2 \right) \right]}^{2}}\left[ q\left( 2 \right) \right]+{{\left[ p\left( 3 \right) \right]}^{2}}\left[ q\left( 3 \right) \right]+{{\left[ p\left( 4 \right) \right]}^{2}}\left[ q\left( 4 \right) \right]+{{\left[ p\left( 5 \right) \right]}^{2}}\left[ q\left( 5 \right) \right] \\  ={{1}^{2}}\times 4+{{0}^{2}}\times 5+{{5}^{2}}\times 3+{{2}^{2}}\times 1+{{3}^{2}}\times 2+{{4}^{2}}\times 0 \\  =4+0+75+4+18+0  =101$

 There is a function $x\left( t \right)$that is the inverse of $t\left( x \right)$ as shown in the graph below. Also there is other function defined as $y\left( x \right)=\left\{ \begin{align}  & x;x\le 3 \\  & 0;x>4 \\ \end{align} \right.$ $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/78e62c45-24ed-4d37-8089-d05e6b1d7a4b1739127865841461868.png"/>           $ \\ $ Find the value of $\sum\limits_{n=0}^{n=5}{\left[ t\left( n \right) \right]}\left[ y\left( n+2 \right) \right]$

 $x\left[ t \right]$: $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/710463ad-53cd-498f-a6ce-73825a3a367c1861595278064163657.png"/> $ \\ $ $t\left[ n \right]$:$ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/2c3a9d01-48e9-4d93-b0c0-60d9c0aa5ecf7017267010397704830.png"/> $ \\ \sum\limits_{n=0}^{n=5}{\left[ t\left( n \right) \right]}\left[ y\left( n+2 \right) \right] \\  =\left[ t\left( 0 \right) \right]\left[ y\left( 2 \right) \right]+\left[ t\left( 1 \right) \right]\left[ y\left( 3 \right) \right]+\left[ t\left( 2 \right) \right]\left[ y\left( 4 \right) \right]+\left[ t\left( 3 \right) \right]\left[ y\left( 5 \right) \right]+\left[ t\left( 4 \right) \right]\left[ y\left( 6 \right) \right]+\left[ t\left( 5 \right) \right]\left[ y\left( 7 \right) \right] \\  =\left( 5\times 2 \right)+\left( 3\times 3 \right)+\left( 2\times 0 \right)+\left( 4\times 0 \right)+\left( 0\times 0 \right)+\left( 1\times 0 \right) \\  =10+9  =19$

 Find the no. of solutions of $f\left( x \right)=g\left( x \right)$. f(x) is inverse of $\left[ \dfrac{2y+3}{4y+5} \right]$ ; g(x)$=$$\left| 2x+3 \right|$

 Draw the curve for each function using the knowledge of functions. Number of solutions of two curves are the intersecting points of both the curves. Number of solutions means the number of intersecting points of the curve.

Let f(y)$=$x. So, $ \\  x=\dfrac{2y+3}{4y+5} \\  \Rightarrow x(4y+5)=2y+3 \\  \Rightarrow 4xy+5x=2y+3 \\  \Rightarrow 5x-3=2y-4xy \\ \Rightarrow \dfrac{5x-3}{2-4x}=y=f\left( x \right) \\ $ $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/3e01c1ad-5aed-4dd2-89ae-58ae669405dd6157394067056893436.png"/> $ \\ $ There is one intersection point therefore, the number of solutions is $1$.

 Find the no. of solutions of $f\left( x \right)=g\left( x \right)$ $ \\ $ $f(x)=$ inverse of $\left[ \dfrac{2y+3}{4y+5} \right]$ ; g(x)=$\sin x$

 Draw the curve for each function using the knowledge of functions. Solution of two curves are the intersecting points of both the curves. Number of solutions means the number of intersecting points of the curve.

Let f(y)$=$x. So, $ \\ x=\dfrac{2y+3}{4y+5} \\  \Rightarrow x(4y+5)=2y+3 \\  \Rightarrow 4xy+5x=2y+3 \\  \Rightarrow 5x-3=2y-4xy \\  \Rightarrow \dfrac{5x-3}{2-4x}=y=f\left( x \right) \\ $ $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/079f23f6-7e47-412c-ab76-ca194c162de85410284826312504142.png"/> $ \\ $ It will have one intersection point thus one no. of solution.

 Find the inverse of the following diagram $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/aafb8612-8a2f-4ec5-bde4-3a9a564d1a117443462359499840260.png"/>

 Check the graph at which the co – ordinates of the given graph and the sign of asymptotes of the given graph are reversed to get the $r=$ inverse function of the given graph.

The $x,y$ co – ordinates of a given function will get reversed for the inverse function. $ \\ $ For a given graph, the $'x'$ asymptote is positive and $'y'$ asymptote is negative which will get reversed for the inverse function. $ \\ $ The both above points are seen only in option $(a)$.