monotone f(x)= x/(x^2+1)

{ "query": { "display": "monotone intervals $$f\\left(x\\right)=\\frac{x}{x^{2}+1}$$", "symbolab_question": "FUNCTION#monotone f(x)=\\frac{x}{x^{2}+1}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "monotone", "default": "\\mathrm{Decreasing}:-\\infty <x<-1,\\mathrm{Increasing}:-1<x<1,\\mathrm{Decreasing}:1<x<\\infty ", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Monotone Intervals of $$\\frac{x}{x^{2}+1}:{\\quad}$$Decreasing$$:-\\infty\\:<x<-1,\\:$$Increasing$$:-1<x<1,\\:$$Decreasing$$:1<x<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Monotone intervals definition", "text": "If $$f'\\left(x\\right)>0\\:$$then $$f\\left(x\\right)\\:$$is increasing.<br/>If $$f'\\left(x\\right)<0\\:$$then $$f\\left(x\\right)\\:$$is decreasing." }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)=\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{x}{x^{2}+1}\\right)", "steps": [ { "type": "step", "primary": "Apply the Quotient Rule: $$\\left(\\frac{f}{g}\\right)'=\\frac{f'{\\cdot}g-g'{\\cdot}f}{g^{2}}$$", "result": "=\\frac{\\frac{dx}{dx}\\left(x^{2}+1\\right)-\\frac{d}{dx}\\left(x^{2}+1\\right)x}{\\left(x^{2}+1\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}+1\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)+\\frac{d}{dx}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\left(1\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "type": "step", "result": "=2x+0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1\\cdot\\:\\left(x^{2}+1\\right)-2xx}{\\left(x^{2}+1\\right)^{2}}" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(x^{2}+1\\right)-2xx=-x^{2}+1$$", "input": "1\\cdot\\:\\left(x^{2}+1\\right)-2xx", "result": "=\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$1\\cdot\\:\\left(x^{2}+1\\right)=x^{2}+1$$", "input": "1\\cdot\\:\\left(x^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(x^{2}+1\\right)=\\left(x^{2}+1\\right)$$", "result": "=\\left(x^{2}+1\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=x^{2}+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HNREWIJAGttVJ8tLuFhXKz2gYtQEl4V79ys4pDeUXjWjkVi15I8rBefLi4Iyt2wrZfipvhu260EBw0vyu63vxk1KmoOKsbKJCeuDsUvATG/wKYd15lciiD0rY3SmLhdAsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "$$2xx=2x^{2}$$", "input": "2xx", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$xx=\\:x^{1+1}$$" ], "result": "=2x^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74UAakB14Lbm7SHYIpLTwTnCQoYlYQ8U+Tfyx0kyzI8iSVveXeWzQO/GlTVao5UKXszTt6qIJZczvODM49/dKgo8BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "step", "result": "=x^{2}+1-2x^{2}" }, { "type": "step", "primary": "Group like terms", "result": "=x^{2}-2x^{2}+1" }, { "type": "step", "primary": "Add similar elements: $$x^{2}-2x^{2}=-x^{2}$$", "result": "=-x^{2}+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HNREWIJAGttVJ8tLuFhXK54kF2IpnZAO71yKpjphLq8JQJZuTAY5js+oqjdT8ksl83iFzgC1jRav6XSHifJOjYpiDBqHg2RlromhyTcRK5hXdYO2Wwjmwk/VR33gpcrJoUMwwlVgJk5w51zpcjzcyA==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Decreasing$$:-\\infty\\:<x<-1,\\:$$Increasing$$:-1<x<1,\\:$$Decreasing$$:1<x<\\infty\\:$$", "input": "f\\:{^{\\prime}}\\left(x\\right)=\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)>0:{\\quad}-1<x<1$$", "input": "\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}>0", "steps": [ { "type": "interim", "title": "Factor $$\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}:{\\quad}\\frac{-\\left(x+1\\right)\\left(x-1\\right)}{\\left(x^{2}+1\\right)^{2}}$$", "input": "\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}", "result": "\\frac{-\\left(x+1\\right)\\left(x-1\\right)}{\\left(x^{2}+1\\right)^{2}}>0", "steps": [ { "type": "interim", "title": "Factor $$-x^{2}+1:{\\quad}-\\left(x+1\\right)\\left(x-1\\right)$$", "input": "-x^{2}+1", "result": "=\\frac{-\\left(x+1\\right)\\left(x-1\\right)}{\\left(x^{2}+1\\right)^{2}}", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(x^{2}-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$x^{2}-1:{\\quad}\\left(x+1\\right)\\left(x-1\\right)$$", "input": "x^{2}-1", "steps": [ { "type": "step", "primary": "Rewrite $$1$$ as $$1^{2}$$", "result": "=x^{2}-1^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$x^{2}-1^{2}=\\left(x+1\\right)\\left(x-1\\right)$$" ], "result": "=\\left(x+1\\right)\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-\\left(x+1\\right)\\left(x-1\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Multiply both sides by $$-1$$ (reverse the inequality)", "result": "\\frac{\\left(-\\left(x+1\\right)\\left(x-1\\right)\\right)\\left(-1\\right)}{\\left(x^{2}+1\\right)^{2}}<0\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\frac{\\left(x+1\\right)\\left(x-1\\right)}{\\left(x^{2}+1\\right)^{2}}<0", "meta": { "solvingClass": "Solver" } }, { "type": "interim", "title": "Identify the intervals", "result": "-1<x<1", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$\\frac{\\left(x+1\\right)\\left(x-1\\right)}{\\left(x^{2}+1\\right)^{2}}$$" }, { "type": "interim", "title": "Find the signs of $$x+1$$", "steps": [ { "type": "interim", "title": "$$x+1=0:{\\quad}x=-1$$", "input": "x+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x+1=0", "result": "x=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "x=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x+1<0:{\\quad}x<-1$$", "input": "x+1<0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x+1<0", "result": "x<-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x+1-1<0-1" }, { "type": "step", "primary": "Simplify", "result": "x<-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MBLwY1ccHHx+0uv6QeFtmpN1pXT08zEQpn0WJ6CFMXD+Bj/OTEQM9+GhM5gnqbimIv3pLYfKyQbKCshhkaZHUUdyPEhiYjGR8ysyUCIfCYSyiHU3VsxZ0Jvlgz1Dh/2W0zIpeqOKDWh7ydUbxgKhdXr3g+Hzr2XIOhBR2yznfmuN7M43BKNThiZyY+QXK4n7qUXmVRy7ARH6c5LXg5AdioHME7TtYugFK35ChD6AUs/Cdt8MO2Xl11SUnMP3Qefr+iMddZHikSDFcuhIgQ9u6eRTlOQwxYRLnXIWxOQlPqV+UHjj5wCFuXTkPs3WjhedhQPEG7O1Z1Z0SY5cF7csulY82rMSGQ/moPqKg1Tiz1+L1z+ccaEiq89Gm211KLaWkmP/nEop0HkUfA7gb98efybEXmnEB6OAwF1d/w0eX6aTd0r2E3qk+TNU1/PySNy3ubwZIBT6UVaOV4ffO4yINiyRt4wIxnMKMEpRk0TkrfewiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x+1>0:{\\quad}x>-1$$", "input": "x+1>0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x+1>0", "result": "x>-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x+1-1>0-1" }, { "type": "step", "primary": "Simplify", "result": "x>-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x-1$$", "steps": [ { "type": "interim", "title": "$$x-1=0:{\\quad}x=1$$", "input": "x-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1=0", "result": "x=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "x=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Jnce9ywdm38UKuWBgTSrMZN1pXT08zEQpn0WJ6CFMXD+Bj/OTEQM9+GhM5gnqbimIv3pLYfKyQbKCshhkaZHUWujmp0NJGTqc2b2PfkAkMeJLbrsy9PUkt1PkbIMRxhVzPywKoVbgpEyLnOm1p35SKygCIJkvFvbYplW0YhTWI0GB82NzCvjutirSS4+6Mo0lcq81LWYZkqKdlKMQed+qs/93sVHhwWy7HsAaDkkE8xJ9H/cBKrAkI41v4vJw/nn8AspaOQmhm7uo64AGDZxV8QVTIXxHbnuqrGIe5Uv+ELDIUjprCA+XIptxfb/EbOOB4CES9nGxD2S5ut+HDkiOxtWKyJwU+3om7eHGIbx8A80UCRHZyRKDzQfKBxnNPRTvJXrhRQTKZrV0QqJ1oeAGfve1FkLGYy7pfNHW7ge8PWtSON6otHROlKM8B29ZaSGBJJRiEsul3reIJEqHjJ5pZRtZJJOUDSa/tucvyzZL8i/Mg94S0N9we//Py6WzxN6" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x-1<0:{\\quad}x<1$$", "input": "x-1<0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1<0", "result": "x<1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1<0+1" }, { "type": "step", "primary": "Simplify", "result": "x<1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x-1>0:{\\quad}x>1$$", "input": "x-1>0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1>0", "result": "x>1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1>0+1" }, { "type": "step", "primary": "Simplify", "result": "x>1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$\\left(x^{2}+1\\right)^{2}$$", "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "step", "primary": "Find singularity points" }, { "type": "interim", "title": "Find the zeros of the denominator $$\\left(x^{2}+1\\right)^{2}:{\\quad}$$No Solution", "input": "\\left(x^{2}+1\\right)^{2}=0", "steps": [ { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$" }, { "type": "interim", "title": "Solve $$x^{2}+1=0:{\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "x^{2}+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x^{2}+1=0", "result": "x^{2}=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x^{2}+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "$$x^{2}$$ cannot be negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Find Denom Zeroes Title 1Eq" } }, { "type": "step", "primary": "Summarize in a table:", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|c|}\\hline &x<-1&x=-1&-1<x<1&x=1&x>1\\\\\\hline x+1&-&0&+&+&+\\\\\\hline x-1&-&-&-&0&+\\\\\\hline (x^{2}+1)^{2}&+&+&+&+&+\\\\\\hline \\frac{(x+1)(x-1)}{(x^{2}+1)^{2}}&+&0&-&0&+\\\\\\hline \\end{array}$$" ] }, { "type": "step", "primary": "Identify the intervals that satisfy the required condition: $$<\\:0$$", "result": "-1<x<1" } ], "meta": { "interimType": "Identify The Intervals NoCol 0Eq" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)<0:{\\quad}x<-1\\lor\\:x>1$$", "input": "\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}<0", "steps": [ { "type": "interim", "title": "Factor $$\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}:{\\quad}\\frac{-\\left(x+1\\right)\\left(x-1\\right)}{\\left(x^{2}+1\\right)^{2}}$$", "input": "\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}", "result": "\\frac{-\\left(x+1\\right)\\left(x-1\\right)}{\\left(x^{2}+1\\right)^{2}}<0", "steps": [ { "type": "interim", "title": "Factor $$-x^{2}+1:{\\quad}-\\left(x+1\\right)\\left(x-1\\right)$$", "input": "-x^{2}+1", "result": "=\\frac{-\\left(x+1\\right)\\left(x-1\\right)}{\\left(x^{2}+1\\right)^{2}}", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(x^{2}-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$x^{2}-1:{\\quad}\\left(x+1\\right)\\left(x-1\\right)$$", "input": "x^{2}-1", "steps": [ { "type": "step", "primary": "Rewrite $$1$$ as $$1^{2}$$", "result": "=x^{2}-1^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$x^{2}-1^{2}=\\left(x+1\\right)\\left(x-1\\right)$$" ], "result": "=\\left(x+1\\right)\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-\\left(x+1\\right)\\left(x-1\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Multiply both sides by $$-1$$ (reverse the inequality)", "result": "\\frac{\\left(-\\left(x+1\\right)\\left(x-1\\right)\\right)\\left(-1\\right)}{\\left(x^{2}+1\\right)^{2}}>0\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\frac{\\left(x+1\\right)\\left(x-1\\right)}{\\left(x^{2}+1\\right)^{2}}>0", "meta": { "solvingClass": "Solver" } }, { "type": "interim", "title": "Identify the intervals", "result": "x<-1\\lor\\:x>1", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$\\frac{\\left(x+1\\right)\\left(x-1\\right)}{\\left(x^{2}+1\\right)^{2}}$$" }, { "type": "interim", "title": "Find the signs of $$x+1$$", "steps": [ { "type": "interim", "title": "$$x+1=0:{\\quad}x=-1$$", "input": "x+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x+1=0", "result": "x=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "x=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x+1<0:{\\quad}x<-1$$", "input": "x+1<0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x+1<0", "result": "x<-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x+1-1<0-1" }, { "type": "step", "primary": "Simplify", "result": "x<-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x+1>0:{\\quad}x>-1$$", "input": "x+1>0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x+1>0", "result": "x>-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x+1-1>0-1" }, { "type": "step", "primary": "Simplify", "result": "x>-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x-1$$", "steps": [ { "type": "interim", "title": "$$x-1=0:{\\quad}x=1$$", "input": "x-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1=0", "result": "x=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "x=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x-1<0:{\\quad}x<1$$", "input": "x-1<0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1<0", "result": "x<1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1<0+1" }, { "type": "step", "primary": "Simplify", "result": "x<1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x-1>0:{\\quad}x>1$$", "input": "x-1>0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1>0", "result": "x>1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1>0+1" }, { "type": "step", "primary": "Simplify", "result": "x>1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$\\left(x^{2}+1\\right)^{2}$$", "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "step", "primary": "Find singularity points" }, { "type": "interim", "title": "Find the zeros of the denominator $$\\left(x^{2}+1\\right)^{2}:{\\quad}$$No Solution", "input": "\\left(x^{2}+1\\right)^{2}=0", "steps": [ { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$" }, { "type": "interim", "title": "Solve $$x^{2}+1=0:{\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "x^{2}+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x^{2}+1=0", "result": "x^{2}=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x^{2}+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fyAweyQvVwJsSQQKDjm97GBUecqxlYikFkylsRQfVW+rpQiLwlAzinHtmVuyD/tLS8DfmRBUqpCWOuo05WrN0LyqZizgxqQXqQv8DcRaGZQEpZhSgqdW5LMAmuoih+kWcXGfWuywtjebrMbja5E8JJDDsPvhQkQSaykqx9sc+sZsadU8wkw3RDM77lql3QfxXpaticrwizP+rml+++I/W9OY4K9o7f7CTm9zs50BJVhNMtlvIkfP+RsK/nN3S+3fl8sqe9+ZFpM5Ptn8RQOTf0gYHD6GTafv5uuph0ZRztun6WU7ybUlSwezFhLKvkKNVvuqUoYkKmBK6fSR2cNlgkfqfT3HSxldmNt6rUcfvySCyrAO9zMQDZyjEiWM7qI5HNLamESyiXkEVNU4Be3yXlX3rP7J381OCLdf7C7WWRDvbBmbuQNTF0TphKZ8Ruvas13fHF95iR4QWD/SCcnUfFHyA5GEFWW3LlP4nBK806VetYVYdLe45VxLoQDaQoEh" } }, { "type": "step", "primary": "$$x^{2}$$ cannot be negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Find Denom Zeroes Title 1Eq" } }, { "type": "step", "primary": "Summarize in a table:", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|c|}\\hline &x<-1&x=-1&-1<x<1&x=1&x>1\\\\\\hline x+1&-&0&+&+&+\\\\\\hline x-1&-&-&-&0&+\\\\\\hline (x^{2}+1)^{2}&+&+&+&+&+\\\\\\hline \\frac{(x+1)(x-1)}{(x^{2}+1)^{2}}&+&0&-&0&+\\\\\\hline \\end{array}$$" ] }, { "type": "step", "primary": "Identify the intervals that satisfy the required condition: $$>\\:0$$", "result": "x<-1\\lor\\:x>1" } ], "meta": { "interimType": "Identify The Intervals NoCol 0Eq" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "Combine intervals with domain", "result": "-\\infty\\:<x<-1,\\:-1<x<1,\\:1<x<\\infty\\:", "steps": [ { "type": "interim", "title": "Domain of $$\\frac{x}{x^{2}+1}\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "step", "primary": "The function has no undefined points nor domain constraints. Therefore, the domain is", "result": "-\\infty\\:<x<\\infty\\:" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "interim", "title": "Combine $$-1<x<1\\:$$ with domain:$${\\quad}-1<x<1$$", "input": "-1<x<1\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "-1<x<1" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$-\\infty\\:<x<-1\\:$$ with domain:$${\\quad}-\\infty\\:<x<-1$$", "input": "-\\infty\\:<x<-1\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "-\\infty\\:<x<-1" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$1<x<\\infty\\:\\:$$ with domain:$${\\quad}1<x<\\infty\\:$$", "input": "1<x<\\infty\\:\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "1<x<\\infty\\:" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "step", "result": "-\\infty\\:<x<-1,\\:-1<x<1,\\:1<x<\\infty\\:" } ], "meta": { "interimType": "Combine Intervals With Domain 0Eq" } }, { "type": "step", "primary": "Summary of the monotone intervals behavior", "secondary": [ "$$\\begin{array}{|c|c|c|c|}\\hline &-\\infty <x<-1&-1<x<1&1<x<\\infty \\\\\\hline \\mathrm{Sign}&f {^{\\prime}}(x)<0&f {^{\\prime}}(x)>0&f {^{\\prime}}(x)<0\\\\\\hline \\mathrm{Behavior}&\\mathrm{Decreasing}&\\mathrm{Increasing}&\\mathrm{Decreasing}\\\\\\hline \\end{array}$$" ] }, { "type": "step", "result": "\\mathrm{Decreasing}:-\\infty\\:<x<-1,\\:\\mathrm{Increasing}:-1<x<1,\\:\\mathrm{Decreasing}:1<x<\\infty\\:" } ], "meta": { "interimType": "Function Find Intervals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGUL8PhNw/UnccDjDvkOKjBsievSRG8FbtJ67MZcqchnBIfFLvhIw/699NIWZ6gDQnSGLMy6w719XZab5nwoBBPwno630PkfrCP0c0TyjW7F2GAt9iheQQcR/4RWGi+OkPaTtwLOQ1U39p54Is2KoSLsN/Pi2+45wlh/W3uZ+DwusL6Rzp9TUMFq2tBBPiGww" } }, { "type": "step", "result": "\\mathrm{Decreasing}:-\\infty\\:<x<-1,\\:\\mathrm{Increasing}:-1<x<1,\\:\\mathrm{Decreasing}:1<x<\\infty\\:" } ], "meta": { "solvingClass": "Function Monotone" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{x}{x^{2}+1}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }