asymptotes of f(x)=(4x)/(x-3)

{ "query": { "display": "asymptotes $$f\\left(x\\right)=\\frac{4x}{x-3}$$", "symbolab_question": "FUNCTION#asymptotes f(x)=\\frac{4x}{x-3}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "asymptotes", "default": "\\mathrm{Vertical}: x=3,\\mathrm{Horizontal}: y=4", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Asymptotes of $$\\frac{4x}{x-3}:\\quad\\:$$Vertical$$:\\:x=3,\\:$$Horizontal$$:\\:y=4$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$\\frac{4x}{x-3}:{\\quad}x=3$$", "input": "\\frac{4x}{x-3}", "steps": [ { "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:$${\\quad}x=3$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{4x}{x-3}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x-3=0:{\\quad}x=3$$", "input": "x-3=0", "steps": [ { "type": "interim", "title": "Move $$3\\:$$to the right side", "input": "x-3=0", "result": "x=3", "steps": [ { "type": "step", "primary": "Add $$3$$ to both sides", "result": "x-3+3=0+3" }, { "type": "step", "primary": "Simplify", "result": "x=3" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=3" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "The vertical asymptotes are:", "result": "x=3" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5dVnUL0bpHaJzHH2MFgu3oeKtWmUUIhY05nZ+7Z/FKmsDeqXxdc+rps1CUyb7fqI2Ge3y3MwMsDZJeby9OVaCdILBuGK/MPjtR4jogh8jhqHIXgVVtNqbKMgpXhAz5x1mPA==" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$\\frac{4x}{x-3}:{\\quad}y=4$$", "input": "\\frac{4x}{x-3}", "steps": [ { "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$$=1.\\:$$The degree of the denominator$$=1$$", "secondary": [ "The degrees are equal, the horizontal asymptote is: $$y=\\frac{\\mathrm{numerator's\\:leading\\:coefficient}}{\\mathrm{denominator's\\:leading\\:coefficient}}$$" ] }, { "type": "step", "primary": "Numerator's leading coefficient$$=4,\\:$$Denominator's leading coefficient$$=1$$", "result": "y=\\frac{4}{1}" }, { "type": "step", "primary": "The horizontal asymptote is:", "result": "y=4" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0PihooH8V6rf2M5+iqpeDLe57MtoReGuVLNq6uBOi7XYGoz/8LfSxJ+0AgVLpCSnLX0iSlRKte6hHzCtOdADy1lL+sOsEuKHoiRQqchnbg8AZByYudQ5wBXVSTF8ZZuUXLYGag==" } }, { "type": "interim", "title": "Slant Asymptotes of $$\\frac{4x}{x-3}:{\\quad}$$None", "input": "\\frac{4x}{x-3}", "steps": [ { "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$$=1.\\:$$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/tmv96KFBl8JLzkMyoq4+quL72YQhpYDqplqKYbd/PJFsARGNwg8YaTQqXIsg4ao8OzbHx2uqecb/OAq5ZJK+9mGRjY15dzXRL5/mz2Y9FubCin39FV2XjVNEekh8JtG3ZXDlTQQ==" } }, { "type": "step", "result": "\\mathrm{Vertical}:\\:x=3,\\:\\mathrm{Horizontal}:\\:y=4" } ], "meta": { "solvingClass": "Function Asymptotes" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{4x}{x-3}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }