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$$\\\\ \\text{The domain of the given function is }\\left[ -1,0 \\right]\\cup \\left[ 1,\\infty \\right) \\\\ \\text{The roots of the function }{{x}^{3}}-x=0\\text{ are }x=-1, x=0, x=1 \\\\ \\text{Which means the value of “y” is “0” at these three values of “x”} \\\\ \\text{Since the function is given with a negative square root, the values of “y” will be negative.}\\\\ \\text{To get the local maxima or minima of } y=-\\sqrt{{{x}^{3}}-x}, \\dfrac{dy}{dx}=0. \\\\ \\Rightarrow \\dfrac{1-3{{x}^{2}}}{2\\sqrt{{{x}^{3}}-x}}=0\\Leftrightarrow x=\\dfrac{-1}{\\sqrt{3}} \\text{ following the domain of “y”.} \\\\ \\text{Since }{{x}^{3}} > x\\text{ in the interval }\\left( 1,\\infty \\right), \\\\ \\text{It will not have any local minimum in the interval.} \\\\ \\text{Graphs can be drawn by checking the values of “y” at different values of “x”.}$$","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Graphing square root functions Quiz 1","text":" Which of the following graphs represent the given square root function $$y=\\sqrt{-{{x}^{4}}+{{x}^{3}}}$$?","comment":{"@type":"Comment","text":" Check the domain and the values of $x$ at which local extrema occur to get the required graph."},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" Not a valid function","position":0},{"@type":"Answer","encodingFormat":"text/html","text":"
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$$\\\\ \\text{Domain of the function is }\\left[ 0,1 \\right] \\\\ \\text{Using the property that }f\\left( x \\right)\\ge 0\\text{ for }\\sqrt{f\\left( x \\right)}\\text{ to be valid.} \\\\ \\text{For local extremum }\\\\ \\dfrac{dy}{dx}=0\\Leftrightarrow \\dfrac{-4{{x}^{3}}+3{{x}^{2}}}{\\sqrt{-{{x}^{4}}+{{x}^{3}}}}=0\\Leftrightarrow x=0,\\dfrac{3}{4} \\\\ \\text{ Which are in the domain and the values of “y” is 0 at }x=0\\text{ and }x=1\\text{ to get the required graph.}$$","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Graphing square root functions Quiz 1","text":" Plot the graph of the function: $$y=\\left\\{ \\begin{matrix}1+\\sqrt{\\left| x-1 \\right|},\\text{ for }x < 1 \\\\-\\text{sgn} \\left( x-1 \\right),\\text{ otherwise} \\\\\\end{matrix} \\right.$$","comment":{"@type":"Comment","text":" Use the property of modulus function and signum function followed by the shifting property of plotting square root functions to get the required plot."},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":"
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","position":3,"answerExplanation":{"@type":"Comment","text":"
$$\\\\ \\text{Making use of the property of modulus function} \\left | x - a \\right | = \\left\\{\\begin{matrix} x-a, & \\text{for } x > a\\\\ -(x-a), & \\text{for } x < a\\\\ 0, & \\text{for } x = a \\end{matrix}\\right.\\\\ \\text{And signum function } sgn(x-a) = \\left\\{\\begin{matrix} 1, & \\text{for } x > a\\\\ -1, & \\text{for } x < a\\\\ 0, & \\text{for } x = a \\end{matrix}\\right. \\\\ \\text{Which makes the square root function}, y = \\left\\{\\begin{matrix} 1+\\sqrt{-(x-1)}, & \\text{for } x < 1\\\\ 0, & \\text{for } x = 1\\\\ -1, & \\text{for } x > 1\\end{matrix}\\right. \\Leftrightarrow y = \\left\\{\\begin{matrix} 1+\\sqrt{1-x}, & \\text{for } x < 1\\\\ 0, & \\text{for } x = 1\\\\ -1, & \\text{for } x > 1 \\end{matrix}\\right. \\\\ \\text{Plotting the obtained function by using the shifting property of square root function in the interval} \\\\ \\text{and showing the discontinuity at } $$","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Graphing square root functions Quiz 1","text":" Which of the following graphs represent the given square root function: $$y=\\sqrt{{{e}^{\\dfrac{1}{\\left| x-1 \\right|}}}}-1$$?","comment":{"@type":"Comment","text":" Check the asymptotes of the given functions and plot it at those points. Shifting of the function along the y-axis due to subtraction of 1 from the square root function for the required plot."},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":"
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$$\\\\ \\text{Making use of the property that } \\dfrac{1}{\\left| x-a \\right|}\\text{ has an asymptote at x=a }\\\\ \\text{Which tells that the given function has asymptote at x=1} \\\\ \\text{And the property that the minimum value of }{{e}^{\\dfrac{1}{\\left| x-a \\right|}}}\\text{ is 1 at }\\left| x-a \\right|=\\infty (\\text{which means that }{{e}^{\\dfrac{1}{\\left| x-1 \\right|}}}\\text{ is asymptote to y=1)}. \\\\ \\text{Shifting of the function downward later due to subtraction of -1 from the }\\sqrt{{{e}^{\\dfrac{1}{\\left| x-1 \\right|}}}}.$$.","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Graphing square root functions Quiz 1","text":" Plot the given square root function: $$y=1-\\sqrt{\\log x-1}$$?","comment":{"@type":"Comment","text":" Finding the domain of the function and then plotting the function $y=-\\sqrt{\\log x-1}$ by checking values at some of the values of $x$. Using the shifting property of square root functions to get the plot of $y=1-\\sqrt{\\log x-1}$."},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":"
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$$\\\\ \\text{Domain of the function }y=1-\\sqrt{\\log x-1}\\text{ is }\\left[ e,\\infty \\right)\\text{ as the function }\\sqrt{f\\left( x \\right)}\\text{ is valid only if }f\\left( x \\right)\\ge 0. \\\\ \\text{ Plotting the graph of the function }y=-\\sqrt{\\log x-1}\\text{ by checking its values at x=e and other values of “x”.} \\\\ \\text{Shifting the plotted by 1 unit upwards as the function } y=-\\sqrt{\\log x-1}\\text{ is added with 1.}$$","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]}]}