inverse of f(x)=2sqrt(x-5)

{ "query": { "display": "inverse $$f\\left(x\\right)=2\\sqrt{x-5}$$", "symbolab_question": "FUNCTION#inverse f(x)=2\\sqrt{x-5}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\frac{x^{2}}{4}+5", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$2\\sqrt{x-5}:{\\quad}\\frac{x^{2}}{4}+5$$", "steps": [ { "type": "definition", "title": "Function Inverse definition", "text": "A function g is the inverse of function f if for $$y=f\\left(x\\right),\\:\\:x=g\\left(y\\right)\\:$$" }, { "type": "step", "result": "y=2\\sqrt{x-5}" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=2\\sqrt{x-5}", "result": "x=2\\sqrt{y-5}", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=2\\sqrt{y-5}" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZWFgdof0mln5sAB+dS/7ijCD0yqNoTWLwGuqxO0C0cFVz5Lb2DzoR+2lK1ma7A4yFw7HH89RUTaeN3FUkEAdGm+LD3IXcw26W+fH/nWE+8Jg6uvWjigmDRR62v/uo5/c/6/iRbMPfLQeIqHWmDQDbe8=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=2\\sqrt{y-5}$$", "input": "x=2\\sqrt{y-5}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "2\\sqrt{y-5}=x" }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2\\sqrt{y-5}=x", "result": "\\sqrt{y-5}=\\frac{x}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2\\sqrt{y-5}}{2}=\\frac{x}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt{y-5}=\\frac{x}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Square both sides:$${\\quad}y-5=\\frac{x^{2}}{4}$$", "input": "\\sqrt{y-5}=\\frac{x}{2}", "result": "y-5=\\frac{x^{2}}{4}", "steps": [ { "type": "step", "result": "\\left(\\sqrt{y-5}\\right)^{2}=\\left(\\frac{x}{2}\\right)^{2}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt{y-5}\\right)^{2}:{\\quad}y-5$$", "input": "\\left(\\sqrt{y-5}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(y-5\\right)^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=\\left(y-5\\right)^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=y-5", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vz0OwVmz7FbkxdUH+eNAiYGUkcvOaxZAUm5h45yL0emJ9HQfN3qOfAB4kBd9UOFDpA7clkQOa7zoZEIDggSG4WRLd2VwIqlBNByF6663syTD7/TDV+MdcR0GdN6+1FDtJnWywhQ4lVjK8aMqVNSmlYmpXFf3SOUx+H18qfp3MLg=" } }, { "type": "interim", "title": "Expand $$\\left(\\frac{x}{2}\\right)^{2}:{\\quad}\\frac{x^{2}}{4}$$", "input": "\\left(\\frac{x}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{x^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\frac{x^{2}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PYIowdXt2t1vjFLpvo+g6Gez183nLWWeYpvlsOrOijfptRYmj6WUpgaKrNIresW2wNmnTNPfSXaFwWcdWyIh039H/SFX0n2Ge344m0fgESR0U1L0UOfZXYZXOhi7eOgpx3URRsP3IGXaYScsUlze+fotFMd5aa71+dAhX5ea6wE=" } }, { "type": "step", "result": "y-5=\\frac{x^{2}}{4}" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDaZCNCMlwFw9Zrpnfgo8QW/WOwxW0IR0wYYVPLXyXeRYXflcl9kRnnQk+MFsuik2kQ1Bwl6I+B+YRPfML31vW1la1ygC+3Kr62ZdG5LvfiraN6Hv6MoTMtvtU0IQwXdn/oQmkZLMOYzj88wQWB+gSMtS4WPVolS0Rr7K51IRWwnQ==" } }, { "type": "interim", "title": "Solve $$y-5=\\frac{x^{2}}{4}:{\\quad}y=\\frac{x^{2}}{4}+5$$", "input": "y-5=\\frac{x^{2}}{4}", "steps": [ { "type": "interim", "title": "Move $$5\\:$$to the right side", "input": "y-5=\\frac{x^{2}}{4}", "result": "y=\\frac{x^{2}}{4}+5", "steps": [ { "type": "step", "primary": "Add $$5$$ to both sides", "result": "y-5+5=\\frac{x^{2}}{4}+5" }, { "type": "step", "primary": "Simplify", "result": "y=\\frac{x^{2}}{4}+5" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "y=\\frac{x^{2}}{4}+5" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "\\frac{x^{2}}{4}+5" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2\\sqrt{x-5}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }