parity (tan(2x))/x

{ "query": { "display": "parity $$\\frac{\\tan\\left(2x\\right)}{x}$$", "symbolab_question": "FUNCTION#parity \\frac{\\tan(2x)}{x}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "parity", "default": "\\mathrm{Even}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Parity of $$\\frac{\\tan\\left(2x\\right)}{x}:{\\quad}$$Even", "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": "$$f\\left(-x\\right):{\\quad}\\frac{\\tan\\left(2x\\right)}{x}$$", "steps": [ { "type": "step", "primary": "Plug $$-x\\:$$into $$\\frac{\\tan\\left(2x\\right)}{x}$$", "result": "=\\frac{\\tan\\left(2\\left(-x\\right)\\right)}{-x}" }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{\\tan\\left(-2x\\right)}{-x}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\tan\\left(-2x\\right)}{x}" }, { "type": "step", "primary": "Use the negative angle identity: $$\\tan\\left(-x\\right)=-\\tan\\left(x\\right)$$", "secondary": [], "result": "=-\\frac{-\\tan\\left(2x\\right)}{x}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\left(-\\frac{\\tan\\left(2x\\right)}{x}\\right)" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{\\tan\\left(2x\\right)}{x}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$-f\\left(x\\right):{\\quad}-\\frac{\\tan\\left(2x\\right)}{x}$$", "steps": [ { "type": "step", "primary": "Negate $$\\frac{\\tan\\left(2x\\right)}{x}$$", "result": "=-\\frac{\\tan\\left(2x\\right)}{x}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Check parity of $$\\frac{\\tan\\left(2x\\right)}{x}$$" }, { "type": "interim", "title": "Check if Even:$${\\quad}$$True", "steps": [ { "type": "step", "result": "f\\left(x\\right)=\\frac{\\tan\\left(2x\\right)}{x},\\:f\\left(-x\\right)=\\frac{\\tan\\left(2x\\right)}{x}" }, { "type": "step", "result": "f\\left(x\\right)=f\\left(-x\\right)" }, { "type": "step", "primary": "Therefore", "result": "\\frac{\\tan\\left(2x\\right)}{x}\\mathrm{\\:is\\: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{\\tan\\left(2x\\right)}{x},\\:f\\left(-x\\right)=\\frac{\\tan\\left(2x\\right)}{x}" }, { "type": "step", "result": "-f\\left(x\\right)\\ne\\:f\\left(-x\\right)" }, { "type": "step", "primary": "Therefore", "result": "\\frac{\\tan\\left(2x\\right)}{x}\\mathrm{\\:is\\:not\\:an\\:odd\\:function}" } ], "meta": { "interimType": "Parity Check Odd Title 0Eq" } }, { "type": "step", "result": "\\mathrm{Even}" } ], "meta": { "solvingClass": "Function Parity" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{\\tan(2x)}{x}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }