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This is one of our many quizzes on graphing square root functions. Click on this to access more such competitive quizzes on squares and square roots.

Graphing square root functions Quiz 2

mathematics

 How do you plot the given square root function $y=1+\sqrt{1-{{\tan }^{2}}\left( \dfrac{x}{4} \right)}$

 Check the domain of the function and the values of the function at the values of $x$ in the domain. Shifting property of the square root function is then used to get the required plot.

  <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/bd5f2e19-4cd5-4bed-a3d8-0dc636a90c886696943423081569982.png"/>$\\ \text{Domain of the given function can be found by applying the property that }f\left( x \right)\ge 0\text{ for the function }\sqrt{f\left( x \right)}\text{ to be valid which is }\left[ \left( 4n-1 \right)\pi ,\left( 4n+1 \right)\pi  \right], n\in Z.\\ $ Values at the extremum and mid-values of the interval are found and these are plotted on the graph to get the plot of $y=\sqrt{1-{{\tan }^{2}}\left( \dfrac{x}{4} \right)}.\\ \text{Obtained plot is then shifted upward by 1 unit as }\sqrt{1-{{\tan }^{2}}\left( \dfrac{x}{4} \right)}\text{ is added with 1 to get the required curve.} $

 Which of the following graph represents the given square root function: $y=\dfrac{1}{\sqrt{{{2}^{\dfrac{x}{4}}}-4}}$?

 Checking the domain of the function at which the given function is valid and asymptote to a value of $x$ to get the required plot.

  <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/a66820cd-932f-446a-82e9-466aaa2dcd609079393375307015211.png"/>$\\ \text{Domain of the given function can be calculated by using the properties that }f\left( x \right)\ge 0\text{ if the function }\sqrt{f\left( x \right)}\text{ to be valid and that }g\left( x \right)\ne 0\text{ if the function }\dfrac{1}{g\left( x \right)}\text{ is valid (function is asymptote for the value of “x” at which }g\left( x \right)=0)\text{ for which the result comes out to be }\left( 8,\infty  \right).$Plotting of the given function at different values of $x$ from the domain is done to get the required graph.

 Which of the following graphs represent the given square root function: $y=1+\sqrt{\log \left( \tan \left( x+\dfrac{\pi }{4} \right) \right)}$?

 Finding the domain of the given function by using the properties of square root and logarithmic function, and then plotting at the values of x from the domain gives the required graph.

 <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/f2661b2d-a4e3-472f-afdb-cbcb0257cc674724715110338733485.png"/>$\\ \text{Domain of the function is found by using the properties that }f\left( x \right)\ge 0\text{ if the function }\sqrt{f\left( x \right)}\text{ to be valid and }g\left( x \right) > 0\text{ for the function }\ln \left( g\left( x \right) \right)\text{ to be valid which results as }\left[ 2n\pi ,2n\pi +\dfrac{\pi }{4} \right).\\ \text{ Plotting the function }y=\sqrt{\log \left( \tan \left( x+\dfrac{\pi }{4} \right) \right)}\text{ is done by finding the values of it at the values of “x” from the obtained domain.} \\ \text{Shifting of the plotted function upward by 1 unit gives the required graph as the function }y=\sqrt{\log \left( \tan \left( x+\dfrac{\pi }{4} \right) \right)}\text{ is added with 1.}$

 Which of the following graphs represent the given square root function: $y=\sqrt{\ln \left( {{\sin }^{-1}}\left( {{x}^{2}}+1 \right) \right)}$?

 Finding the values of $x$ for which the given function is valid which gives the required plot.

 $\text{Values of the function }{{x}^{2}}+1\ge 1.\text{When }{{x}^{2}}+1=1,\text{ the value of }{{\sin }^{-1}}\left( {{x}^{2}}+1 \right)=\dfrac{\pi }{2} < e.\\ \text{This makes the }\ln \left( {{\sin }^{-1}}\left( \left| {{x}^{2}}+1 \right| \right) \right) < 0\text{ which doesn’t satisfy the property  }f\left( x \right)\ge 0\text{ if the function }\sqrt{f\left( x \right)}\text{ to be valid.} $

 Which of the following plots represent the given square root function: $y=\sqrt{\cosh \left( \ln \left( {{\tan }^{-1}}\left( x \right) \right) \right)}$?

 Checking the domain by using the properties of the square root and logarithmic function. Plotting the graph by using the principal solution for the $\arctan \left( x \right)$.

 <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/ba68d650-3fd7-4e0d-8ed5-f4ac92cb11fc4394075354129557204.png"/>$\\ \text{Domain of the function is function by using the properties that }f\left( x \right)\ge 0\text{ if the function }\sqrt{f\left( x \right)}\text{ to be valid and }g\left( x \right) > 0\text{ for the function }\ln \left( g\left( x \right) \right)\text{ to be valid which results as }\left( 0,\infty  \right).\\ $ From the domain, it is clear that the given function has an asymptote at x=0 and plotting the different values of function at different values of $x$ gives the required graph. Considering the principal solution for the ${{\tan }^{-1}}\left( x \right)$.