monotone f(x)=sqrt(x-1)

{ "query": { "display": "monotone intervals $$f\\left(x\\right)=\\sqrt{x-1}$$", "symbolab_question": "FUNCTION#monotone f(x)=\\sqrt{x-1}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "monotone", "default": "\\mathrm{Increasing}:1<x<\\infty ", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Monotone Intervals of $$\\sqrt{x-1}:{\\quad}$$Increasing$$:1<x<\\infty\\:$$", "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": "$$f\\:{^{\\prime}}\\left(x\\right)=\\frac{1}{2\\sqrt{x-1}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x-1}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2\\sqrt{x-1}}\\frac{d}{dx}\\left(x-1\\right)$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x-1}\\right)", "result": "=\\frac{1}{2\\sqrt{x-1}}\\frac{d}{dx}\\left(x-1\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sqrt{u},\\:\\:u=x-1$$" ], "result": "=\\frac{d}{du}\\left(\\sqrt{u}\\right)\\frac{d}{dx}\\left(x-1\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\sqrt{u}\\right)=\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{d}{du}\\left(\\sqrt{u}\\right)", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\frac{d}{du}\\left(u^{\\frac{1}{2}}\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=\\frac{1}{2}u^{\\frac{1}{2}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}u^{\\frac{1}{2}-1}:{\\quad}\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{1}{2}u^{\\frac{1}{2}-1}", "result": "=\\frac{1}{2\\sqrt{u}}", "steps": [ { "type": "interim", "title": "$$u^{\\frac{1}{2}-1}=u^{-\\frac{1}{2}}$$", "input": "u^{\\frac{1}{2}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}-1:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{1}{2}-1", "result": "=u^{-\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=-\\frac{1\\cdot\\:2}{2}+\\frac{1}{2}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1\\cdot\\:2+1}{2}" }, { "type": "interim", "title": "$$-1\\cdot\\:2+1=-1$$", "input": "-1\\cdot\\:2+1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=-2+1" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-2+1=-1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s731snK5z/nd3Sq/6JpCqiX1XTSum/z5kLpMzXS1UJIew02FKSBoQo9V3G05AlnWtTyCE30rzMlUAIVDyhseMBropKGn5MuXZnb2ZCo/hVsBU=" } }, { "type": "step", "result": "=\\frac{-1}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VcI2MpaClJgyGWg1EkySKe0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjMvOxDqXzE3/CFO0TFmffHAH2kDe5DGYTz3TrPquGdIhyukSOA/1RgMKO0TMhInPOMabdUggEogUL9RT7PNKh0VQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{1}{2}u^{-\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$u^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{u}}$$" ], "result": "=\\frac{1}{2}\\cdot\\:\\frac{1}{\\sqrt{u}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{2\\sqrt{u}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\sqrt{u}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JOPQ2g2GS9EQptV8nckZSrH6E/qPf7AlxQDX8MXU5OsAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJv2RkT96g5Q5jVbn5fyeQzwB9pA3uQxmE8906z6rhnSIHimBRYRqHSWeJkuUPhfTC1O468YRFxaQeTFqgRqR2rvsVWktCxa7XSYzIK90x3+aTk5AXTHU+C+TrGKWzqT97A==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{u}}\\frac{d}{dx}\\left(x-1\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-1$$", "result": "=\\frac{1}{2\\sqrt{x-1}}\\frac{d}{dx}\\left(x-1\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjasHTJ4Bo7yMP4qgzIlGESXz3dn1moclLVosWYimSIKdLl7DeVd7l7l/uUT/v1GhNJLZOXSTCXkMFKW90A2Pi/484O9NJ/PDg7/ATYMA16bcvVHNqLba6vmYtJuyr84nSVFJD8NzgqIuC8eLoJx97pls6oM6RhSw9AjzvgS5Bld3SmUvX4+ZxjvBdfdSGCsOwpcB1b+ayqTG1ksTw+VIa6/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x-1\\right)=1$$", "input": "\\frac{d}{dx}\\left(x-1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{dx}{dx}-\\frac{d}{dx}\\left(1\\right)" }, { "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": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "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": "=1-0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{x-1}}\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{2\\sqrt{x-1}}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Increasing$$:1<x<\\infty\\:$$", "input": "f\\:{^{\\prime}}\\left(x\\right)=\\frac{1}{2\\sqrt{x-1}}", "steps": [ { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)>0:{\\quad}x>1$$", "input": "\\frac{1}{2\\sqrt{x-1}}>0", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{1\\cdot\\:2}{2\\sqrt{x-1}}>0\\cdot\\:2" }, { "type": "step", "primary": "Simplify", "result": "\\frac{1}{\\sqrt{x-1}}>0" }, { "type": "step", "primary": "If$$\\quad\\:\\frac{1}{a}>0\\:\\quad$$then$$\\quad\\:a>0$$", "result": "\\sqrt{x-1}>0" }, { "type": "step", "primary": "Square both sides", "result": "\\left(\\sqrt{x-1}\\right)^{2}>0^{2}" }, { "type": "step", "primary": "Simplify", "result": "x-1>0" }, { "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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" } }, { "type": "step", "primary": "Find singularity points" }, { "type": "interim", "title": "Find non-negative values for radicals:$${\\quad}x\\ge\\:1$$", "steps": [ { "type": "step", "primary": "$$\\sqrt{f\\left(x\\right)}\\quad\\Rightarrow\\quad\\:f\\left(x\\right)\\ge{0}\\:$$", "meta": { "general_rule": { "extension": "$$\\sqrt{f\\left(x\\right)}$$ (or any even root) has real values only when $$f\\left(x\\right)\\ge\\:0\\:$$" } } }, { "type": "step", "primary": "For $$\\sqrt{x-1}:{\\quad}x-1\\ge\\:0$$" }, { "type": "interim", "title": "Solve $$x-1\\ge\\:0:{\\quad}x\\ge\\:1$$", "input": "x-1\\ge\\:0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1\\ge\\:0", "result": "x\\ge\\:1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1\\ge\\:0+1" }, { "type": "step", "primary": "Simplify", "result": "x\\ge\\:1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x\\ge\\:1" } ], "meta": { "interimType": "Non Negative Radicals 0Eq" } }, { "type": "step", "primary": "Combine the intervals", "result": "x>1\\land\\:x\\ge\\:1" }, { "type": "interim", "title": "Merge Overlapping Intervals", "input": "x>1\\land\\:x\\ge\\:1", "result": "x>1", "steps": [ { "type": "step", "primary": "The intersection of two intervals is the set of numbers which are in both intervals<br/>$$x>1\\quad$$and$$\\quad\\:x\\ge\\:1$$", "image": "/images/interval?expression=%28y_%7B0%7D%3E1%29%5Cland+%28y_%7B0%7D%5Cge+1%29", "result": "x>1" } ], "meta": { "interimType": "Merge Overlapping Intervals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQPWX96T/qkhrPX+Pdpczvz7+Co0lkJao65xJeqCTwKdh6pfF1z6umzUJTJvt+ojYZJdXexJRlAoShfiCO2R2RDFMuJ8x+I7EWPhppLH+xdnw=" } }, { "type": "step", "primary": "Find singularity points" }, { "type": "interim", "title": "Find non-negative values for radicals:$${\\quad}x\\ge\\:1$$", "steps": [ { "type": "step", "primary": "$$\\sqrt{f\\left(x\\right)}\\quad\\Rightarrow\\quad\\:f\\left(x\\right)\\ge{0}\\:$$", "meta": { "general_rule": { "extension": "$$\\sqrt{f\\left(x\\right)}$$ (or any even root) has real values only when $$f\\left(x\\right)\\ge\\:0\\:$$" } } }, { "type": "step", "primary": "For $$\\sqrt{x-1}:{\\quad}x-1\\ge\\:0$$" }, { "type": "interim", "title": "Solve $$x-1\\ge\\:0:{\\quad}x\\ge\\:1$$", "input": "x-1\\ge\\:0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1\\ge\\:0", "result": "x\\ge\\:1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1\\ge\\:0+1" }, { "type": "step", "primary": "Simplify", "result": "x\\ge\\:1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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"step", "primary": "The intersection of two intervals is the set of numbers which are in both intervals<br/>$$x>1\\quad$$and$$\\quad\\:x\\ge\\:1$$", "image": "/images/interval?expression=%28y_%7B0%7D%3E1%29%5Cland+%28y_%7B0%7D%5Cge+1%29", "result": "x>1" } ], "meta": { "interimType": "Merge Overlapping Intervals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQPWX96T/qkhrPX+Pdpczvz7+Co0lkJao65xJeqCTwKdh6pfF1z6umzUJTJvt+ojYZJdXexJRlAoShfiCO2R2RDFMuJ8x+I7EWPhppLH+xdnw=" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)<0:{\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "\\frac{1}{2\\sqrt{x-1}}<0", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{1\\cdot\\:2}{2\\sqrt{x-1}}<0\\cdot\\:2" }, { "type": "step", "primary": "Simplify", "result": "\\frac{1}{\\sqrt{x-1}}<0" }, { "type": "step", "primary": "If$$\\quad\\:\\frac{1}{a}<0\\:\\quad$$then$$\\quad\\:a<0$$", "result": "\\sqrt{x-1}<0" }, { "type": "step", "primary": "If n is even, $$\\sqrt[n]{u}\\:\\ge\\:0\\:$$for all $$u$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "Combine intervals with domain", "result": "1<x<\\infty\\:", "steps": [ { "type": "interim", "title": "Domain of $$\\sqrt{x-1}\\::{\\quad}x\\ge\\:1$$", "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": "interim", "title": "Find non-negative values for radicals:$${\\quad}x\\ge\\:1$$", "input": "\\sqrt{x-1}", "steps": [ { "type": "step", "primary": "$$\\sqrt{f\\left(x\\right)}\\quad\\Rightarrow\\quad\\:f\\left(x\\right)\\ge{0}\\:$$", "meta": { "general_rule": { "extension": "$$\\sqrt{f\\left(x\\right)}$$ (or any even root) has real values only when $$f\\left(x\\right)\\ge\\:0\\:$$" } } }, { "type": "interim", "title": "Solve $$x-1\\ge\\:0:{\\quad}x\\ge\\:1$$", "input": "x-1\\ge\\:0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1\\ge\\:0", "result": "x\\ge\\:1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1\\ge\\:0+1" }, { "type": "step", "primary": "Simplify", "result": "x\\ge\\:1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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