asymptotes of f(x)=(-5)/(3x+1)

{ "query": { "display": "asymptotes $$f\\left(x\\right)=\\frac{-5}{3x+1}$$", "symbolab_question": "FUNCTION#asymptotes f(x)=\\frac{-5}{3x+1}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "asymptotes", "default": "\\mathrm{Vertical}: x=-\\frac{1}{3},\\mathrm{Horizontal}: y=0", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Asymptotes of $$\\frac{-5}{3x+1}:\\quad\\:$$Vertical$$:\\:x=-\\frac{1}{3},\\:$$Horizontal$$:\\:y=0$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$\\frac{-5}{3x+1}:{\\quad}x=-\\frac{1}{3}$$", "input": "\\frac{-5}{3x+1}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-5}{3x+1}:{\\quad}-\\frac{5}{3x+1}$$", "input": "\\frac{-5}{3x+1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{5}{3x+1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "definition", "title": "Vertical asymptotes of rational Functions", "text": "For rational functions, the vertical asymptotes are the undefined points, also known as the zeros of the denominator, of the simplified function." }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=-\\frac{1}{3}$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$-\\frac{5}{3x+1}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$3x+1=0:{\\quad}x=-\\frac{1}{3}$$", "input": "3x+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "3x+1=0", "result": "3x=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "3x+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "3x=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=-1", "result": "x=-\\frac{1}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{-1}{3}" }, { "type": "step", "primary": "Simplify", "result": "x=-\\frac{1}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=-\\frac{1}{3}" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "The vertical asymptotes are:", "result": "x=-\\frac{1}{3}" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5dVnUL0bpHaJzHH2MFgu3oeL0EyBhDSgLBl+goDywwzMm1sD7NfhsPe7eDHrmjY0mE6hGLWH8mLHPbpUaeRMGiw9pqDA4+Yqrx7lYxZSoPbWUMMYCPK4ogKeDu3PqiFcDbA==" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$\\frac{-5}{3x+1}:{\\quad}y=0$$", "input": "\\frac{-5}{3x+1}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-5}{3x+1}:{\\quad}-\\frac{5}{3x+1}$$", "input": "\\frac{-5}{3x+1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{5}{3x+1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "definition", "title": "Horizontal asymptotes of rational functions", "text": "If denominator's degree > numerator's degree, the x-axis is the horizontal asymptote.<br/>If the degrees are equal, there is an horizontal asymptote: $$y=\\frac{\\mathrm{numerator's\\:leading\\:coefficient}}{\\mathrm{denominator's\\:leading\\:coefficient}}$$<br/>Otherwise, there is no horizontal asymptote." }, { "type": "step", "primary": "The degree of the numerator$$=0.\\:$$The degree of the denominator$$=1$$", "secondary": [ "Denominator's degree > numerator's degree. Therefore, the horizontal asymptote is the x-axis" ] }, { "type": "step", "primary": "The horizontal asymptote is:", "result": "y=0" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0PihooH8V6rf2M5+iqpeDLe57Mt45feoQu9uDtCebfUWV58GgQUxJPyUNnGfVirkcwpVO9CHETsTXzhiqntDPUs2vrMOIBjVD0w41lRvWu8QFmOASQ7ls/U7JFTdOIeezIOObSS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "Slant Asymptotes of $$\\frac{-5}{3x+1}:{\\quad}$$None", "input": "\\frac{-5}{3x+1}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-5}{3x+1}:{\\quad}-\\frac{5}{3x+1}$$", "input": "\\frac{-5}{3x+1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{5}{3x+1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "definition", "title": "Slant asymptotes of rational functions", "text": "If numerator's degree = 1 + denominator's degree, there is a slant asymptote of the form: y=mx+b.<br/>Otherwise there is no slant asymptote" }, { "type": "step", "primary": "The degree of the numerator$$=0.\\:$$The degree of the denominator$$=1$$", "secondary": [ "Numerator's degree $$\\neq\\:$$ 1 + denominator's degree" ] }, { "type": "step", "primary": "Therefore there is no slant asymptote" }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Slant Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMLXiD7VTAsp80tg/tmv96KKbQld7viLBVSBg8HLdc6MW1I+KD8Fcmv6iSz1ZgDey8Hjb2+5NLFZrsH9fcPWg/TflR5YFuAC+s80L+AwOuiAnm+/Lu1Z3FpYbIq7Js4mJzcEOvXhd9/ifrEq2WxtzDjw==" } }, { "type": "step", "result": "\\mathrm{Vertical}:\\:x=-\\frac{1}{3},\\:\\mathrm{Horizontal}:\\:y=0" } ], "meta": { "solvingClass": "Function Asymptotes" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{-5}{3x+1}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }