parity f(x)=(x+4x^3-5)/(5x^3-2x^2+2)

{ "query": { "display": "parity $$f\\left(x\\right)=\\frac{x+4x^{3}-5}{5x^{3}-2x^{2}+2}$$", "symbolab_question": "FUNCTION#parity f(x)=\\frac{x+4x^{3}-5}{5x^{3}-2x^{2}+2}" }, "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 $$\\frac{x+4x^{3}-5}{5x^{3}-2x^{2}+2}:{\\quad}$$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": "$$f\\left(-x\\right):{\\quad}\\frac{-x-4x^{3}-5}{-5x^{3}-2x^{2}+2}$$", "steps": [ { "type": "step", "primary": "Plug $$-x\\:$$into $$\\frac{x+4x^{3}-5}{5x^{3}-2x^{2}+2}$$", "result": "=\\frac{-x+4\\left(-x\\right)^{3}-5}{5\\left(-x\\right)^{3}-2\\left(-x\\right)^{2}+2}" }, { "type": "interim", "title": "$$-x+4\\left(-x\\right)^{3}-5=-x-4x^{3}-5$$", "input": "-x+4\\left(-x\\right)^{3}-5", "steps": [ { "type": "interim", "title": "$$4\\left(-x\\right)^{3}=-4x^{3}$$", "input": "4\\left(-x\\right)^{3}", "steps": [ { "type": "interim", "title": "$$\\left(-x\\right)^{3}=-x^{3}$$", "input": "\\left(-x\\right)^{3}", "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}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZZTfjWGBqBU7pigqgoRJFM0ag8T1MwTer44+aCS/ZFBN+t+mKTpxAGiirL2o+PfBMGHE0EMDeX5PCXJYoIM+miheeGmW50pyvBRBozIzAMU=" } }, { "type": "step", "result": "=4\\left(-x^{3}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-4x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74fuBhOqcqBTaQYA/oCuUz913jtrSFDx+UNsawjlOjV0VLnRXShL19VD1zT/XeJiJHRHdzS5g2HJFanH3nVRLiLHd/RaeJ/vQ2zbUB7tUShqwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=-x-4x^{3}-5" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kLz2Lf96QMdP33TUZ0ENNACWKUbvV6WK3fDUgFtg3Q+iheWTZ5HmsG0deyWm0yDBxLq2z1/rKOKp9QH4/BNsdkO8APOQYmLnANrntkz+70i/X6RAz1g+lAcz8vnXvxb/R+PMUOA/+29XWKe4/0A6AA==" } }, { "type": "step", "result": "=\\frac{-x-4x^{3}-5}{5\\left(-x\\right)^{3}-2\\left(-x\\right)^{2}+2}" }, { "type": "interim", "title": "$$5\\left(-x\\right)^{3}-2\\left(-x\\right)^{2}+2=-5x^{3}-2x^{2}+2$$", "input": "5\\left(-x\\right)^{3}-2\\left(-x\\right)^{2}+2", "steps": [ { "type": "interim", "title": "$$5\\left(-x\\right)^{3}=-5x^{3}$$", "input": "5\\left(-x\\right)^{3}", "steps": [ { "type": "interim", "title": "$$\\left(-x\\right)^{3}=-x^{3}$$", "input": "\\left(-x\\right)^{3}", "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}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZZTfjWGBqBU7pigqgoRJFM0ag8T1MwTer44+aCS/ZFBN+t+mKTpxAGiirL2o+PfBMGHE0EMDeX5PCXJYoIM+miheeGmW50pyvBRBozIzAMU=" } }, { "type": "step", "result": "=5\\left(-x^{3}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-5x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7LK7lsXPS5nTSdnCmsUlJvd13jtrSFDx+UNsawjlOjV0jAJxK5iX6jhXiO5vvUJYJYumNlYD1wFrfO7PZFchQjlLM50OB2Gv3f2hVgH7kX72wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=-5x^{3}-2\\left(-x\\right)^{2}+2" }, { "type": "interim", "title": "$$\\left(-x\\right)^{2}=x^{2}$$", "input": "\\left(-x\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-x\\right)^{2}=x^{2}$$" ], "result": "=x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Hui1Card7t3pVnDuWgo2dM0ag8T1MwTer44+aCS/ZFDeEkwAFPAIXQjZjJNkASgkHtZQiop+i4zHzDWwfTXGQShxCrzCisZpY7aRlDLKd00=" } }, { "type": "step", "result": "=-5x^{3}-2x^{2}+2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7H4s4xx/7P9+wOqsOXXH0zUWIT6ENpTl2+s5F1Se197gJQJZuTAY5js+oqjdT8ksls48idHGWuMh5HvheEGuatYaUhzBKYBvjAmG5MDlpSknKEorz5iYTGRXfzS7zGFvOpXggLSK+o+We3dvkLdzY+ghQlGOEIx/2sZx9ymcZjdo=" } }, { "type": "step", "result": "=\\frac{-x-4x^{3}-5}{-5x^{3}-2x^{2}+2}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$-f\\left(x\\right):{\\quad}-\\frac{x+4x^{3}-5}{5x^{3}-2x^{2}+2}$$", "steps": [ { "type": "step", "primary": "Negate $$\\frac{x+4x^{3}-5}{5x^{3}-2x^{2}+2}$$", "result": "=-\\frac{x+4x^{3}-5}{5x^{3}-2x^{2}+2}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Check parity of $$\\frac{x+4x^{3}-5}{5x^{3}-2x^{2}+2}$$" }, { "type": "interim", "title": "Check if Even:$${\\quad}$$False", "steps": [ { "type": "step", "result": "f\\left(x\\right)=\\frac{x+4x^{3}-5}{5x^{3}-2x^{2}+2},\\:f\\left(-x\\right)=\\frac{-x-4x^{3}-5}{-5x^{3}-2x^{2}+2}" }, { "type": "step", "result": "f\\left(x\\right)\\ne\\:f\\left(-x\\right)" }, { "type": "step", "primary": "Therefore", "result": "\\frac{x+4x^{3}-5}{5x^{3}-2x^{2}+2}\\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+4x^{3}-5}{5x^{3}-2x^{2}+2},\\:f\\left(-x\\right)=\\frac{-x-4x^{3}-5}{-5x^{3}-2x^{2}+2}" }, { "type": "step", "result": "-f\\left(x\\right)\\ne\\:f\\left(-x\\right)" }, { "type": "step", "primary": "Therefore", "result": "\\frac{x+4x^{3}-5}{5x^{3}-2x^{2}+2}\\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+4x^{3}-5}{5x^{3}-2x^{2}+2}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }