parity cot(3)(x)+cot(2)(x)+cot(x)+1

{ "query": { "display": "parity $$\\cot\\left(3\\right)\\left(x\\right)+\\cot\\left(2\\right)\\left(x\\right)+\\cot\\left(x\\right)+1$$", "symbolab_question": "FUNCTION#parity \\cot(3)(x)+\\cot(2)(x)+\\cot(x)+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 $$\\cot\\left(3\\right)x+\\cot\\left(2\\right)x+\\cot\\left(x\\right)+1:{\\quad}$$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": "$$f\\left(-x\\right):{\\quad}1-\\cot\\left(x\\right)-\\cot\\left(2\\right)x-\\cot\\left(3\\right)x$$", "steps": [ { "type": "step", "primary": "Plug $$-x\\:$$into $$\\cot\\left(3\\right)x+\\cot\\left(2\\right)x+\\cot\\left(x\\right)+1$$", "result": "=\\cot\\left(3\\right)\\left(-x\\right)+\\cot\\left(2\\right)\\left(-x\\right)+\\cot\\left(-x\\right)+1" }, { "type": "step", "primary": "Use the negative angle identity: $$\\cot\\left(-x\\right)=-\\cot\\left(x\\right)$$", "secondary": [], "result": "=1-\\cot\\left(x\\right)+\\cot\\left(2\\right)\\left(-x\\right)+\\cot\\left(3\\right)\\left(-x\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=1-\\cot\\left(x\\right)-\\cot\\left(2\\right)x-\\cot\\left(3\\right)x" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$-f\\left(x\\right):{\\quad}-\\cot\\left(3\\right)x-\\cot\\left(2\\right)x-\\cot\\left(x\\right)-1$$", "steps": [ { "type": "step", "primary": "Negate $$\\cot\\left(3\\right)x+\\cot\\left(2\\right)x+\\cot\\left(x\\right)+1$$", "result": "=-\\left(\\cot\\left(3\\right)x+\\cot\\left(2\\right)x+\\cot\\left(x\\right)+1\\right)" }, { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(\\cot\\left(3\\right)x\\right)-\\left(\\cot\\left(2\\right)x\\right)-\\left(\\cot\\left(x\\right)\\right)-\\left(1\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-\\cot\\left(3\\right)x-\\cot\\left(2\\right)x-\\cot\\left(x\\right)-1" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Check parity of $$\\cot\\left(3\\right)x+\\cot\\left(2\\right)x+\\cot\\left(x\\right)+1$$" }, { "type": "interim", "title": "Check if Even:$${\\quad}$$False", "steps": [ { "type": "step", "result": "f\\left(x\\right)=\\cot\\left(3\\right)x+\\cot\\left(2\\right)x+\\cot\\left(x\\right)+1,\\:f\\left(-x\\right)=1-\\cot\\left(x\\right)-\\cot\\left(2\\right)x-\\cot\\left(3\\right)x" }, { "type": "step", "result": "f\\left(x\\right)\\ne\\:f\\left(-x\\right)" }, { "type": "step", "primary": "Therefore", "result": "\\cot\\left(3\\right)x+\\cot\\left(2\\right)x+\\cot\\left(x\\right)+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)=-\\cot\\left(3\\right)x-\\cot\\left(2\\right)x-\\cot\\left(x\\right)-1,\\:f\\left(-x\\right)=1-\\cot\\left(x\\right)-\\cot\\left(2\\right)x-\\cot\\left(3\\right)x" }, { "type": "step", "result": "-f\\left(x\\right)\\ne\\:f\\left(-x\\right)" }, { "type": "step", "primary": "Therefore", "result": "\\cot\\left(3\\right)x+\\cot\\left(2\\right)x+\\cot\\left(x\\right)+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": "\\cot(3)x+\\cot(2)x+\\cot(x)+1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }