parity f(x)= x/(x^3-1)

{ "query": { "display": "parity $$f\\left(x\\right)=\\frac{x}{x^{3}-1}$$", "symbolab_question": "FUNCTION#parity f(x)=\\frac{x}{x^{3}-1}" }, "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 $$\\frac{x}{x^{3}-1}:{\\quad}$$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": "$$f\\left(-x\\right):{\\quad}-\\frac{x}{-x^{3}-1}$$", "steps": [ { "type": "step", "primary": "Plug $$-x\\:$$into $$\\frac{x}{x^{3}-1}$$", "result": "=\\frac{-x}{\\left(-x\\right)^{3}-1}" }, { "type": "interim", "title": "$$\\left(-x\\right)^{3}-1=-x^{3}-1$$", "input": "\\left(-x\\right)^{3}-1", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=-a^{n},\\:$$if $$n$$ is odd", "secondary": [ "$$\\left(-x\\right)^{3}=-x^{3}$$" ], "result": "=-x^{3}-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zdSYCWW4MnLwENrP/lhaviAn9lkDfZkicUGkO3EF+IqKG0G9+Y3O8Yy2ioYnJrz7Or42DoRa4lmvtW+XTBjyKQoBykjCP6j/rkTqTeeYAeoIZg5+s9oUHZWh0PlaTm3O" } }, { "type": "step", "result": "=\\frac{-x}{-x^{3}-1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{x}{-x^{3}-1}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$-f\\left(x\\right):{\\quad}-\\frac{x}{x^{3}-1}$$", "steps": [ { "type": "step", "primary": "Negate $$\\frac{x}{x^{3}-1}$$", "result": "=-\\frac{x}{x^{3}-1}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Check parity of $$\\frac{x}{x^{3}-1}$$" }, { "type": "interim", "title": "Check if Even:$${\\quad}$$False", "steps": [ { "type": "step", "result": "f\\left(x\\right)=\\frac{x}{x^{3}-1},\\:f\\left(-x\\right)=-\\frac{x}{-x^{3}-1}" }, { "type": "step", "result": "f\\left(x\\right)\\ne\\:f\\left(-x\\right)" }, { "type": "step", "primary": "Therefore", "result": "\\frac{x}{x^{3}-1}\\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)=-\\frac{x}{x^{3}-1},\\:f\\left(-x\\right)=-\\frac{x}{-x^{3}-1}" }, { "type": "step", "result": "-f\\left(x\\right)\\ne\\:f\\left(-x\\right)" }, { "type": "step", "primary": "Therefore", "result": "\\frac{x}{x^{3}-1}\\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": "\\frac{x}{x^{3}-1}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }