parity f(x)=sqrt(3x)

{ "query": { "display": "parity $$f\\left(x\\right)=\\sqrt{3x}$$", "symbolab_question": "FUNCTION#parity f(x)=\\sqrt{3x}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "parity", "default": "\\mathrm{Neither\\:even\\:nor\\:odd}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Parity of $$\\sqrt{3x}:{\\quad}$$Neither even nor odd", "steps": [ { "type": "definition", "title": "Parity Definition", "text": "Even Function: A function is even if $$f\\left(-x\\right)=f\\left(x\\right)$$ for all $$x\\in\\mathbb{R}$$<br/>Odd Function: A function is odd if $$f\\left(-x\\right)=-f\\left(x\\right)$$ for all $$x\\in\\mathbb{R}$$" }, { "type": "interim", "title": "$$f\\left(x\\right):{\\quad}\\sqrt{3}\\sqrt{x}$$", "input": "\\sqrt{3x}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\sqrt{3}\\sqrt{x}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UoxHA89ydmh7nJex7umNiiAn9lkDfZkicUGkO3EF+IpIyB7h/T9zVR3i5g4dJo8706qqEqkT4d+n3BFmbi2pRhcXAdW1JEeHPL1Q/CykAHu/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "$$f\\left(-x\\right):{\\quad}\\sqrt{3}\\sqrt{-x}$$", "steps": [ { "type": "step", "primary": "Plug $$-x\\:$$into $$\\sqrt{3}\\sqrt{x}$$", "result": "=\\sqrt{3}\\sqrt{-x}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$-f\\left(x\\right):{\\quad}-\\sqrt{3}\\sqrt{x}$$", "steps": [ { "type": "step", "primary": "Negate $$\\sqrt{3}\\sqrt{x}$$", "result": "=-\\sqrt{3}\\sqrt{x}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Check parity of $$\\sqrt{3}\\sqrt{x}$$" }, { "type": "interim", "title": "Check if Even:$${\\quad}$$False", "steps": [ { "type": "step", "result": "f\\left(x\\right)=\\sqrt{3}\\sqrt{x},\\:f\\left(-x\\right)=\\sqrt{3}\\sqrt{-x}" }, { "type": "step", "result": "f\\left(x\\right)\\ne\\:f\\left(-x\\right)" }, { "type": "step", "primary": "Therefore", "result": "\\sqrt{3}\\sqrt{x}\\mathrm{\\:is\\:not\\:an\\:even\\:function}" } ], "meta": { "interimType": "Parity Check Even Title 0Eq" } }, { "type": "interim", "title": "Check if Odd:$${\\quad}$$False", "steps": [ { "type": "step", "result": "-f\\left(x\\right)=-\\sqrt{3}\\sqrt{x},\\:f\\left(-x\\right)=\\sqrt{3}\\sqrt{-x}" }, { "type": "step", "result": "-f\\left(x\\right)\\ne\\:f\\left(-x\\right)" }, { "type": "step", "primary": "Therefore", "result": "\\sqrt{3}\\sqrt{x}\\mathrm{\\:is\\:not\\:an\\:odd\\:function}" } ], "meta": { "interimType": "Parity Check Odd Title 0Eq" } }, { "type": "step", "result": "\\mathrm{Neither\\:even\\:nor\\:odd}" } ], "meta": { "solvingClass": "Function Parity" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sqrt{3x}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }