asymptotes of (9-3x)/(x-5)

{ "query": { "display": "asymptotes $$\\frac{9-3x}{x-5}$$", "symbolab_question": "FUNCTION#asymptotes \\frac{9-3x}{x-5}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "asymptotes", "default": "\\mathrm{Vertical}: x=5,\\mathrm{Horizontal}: y=-3", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Asymptotes of $$\\frac{9-3x}{x-5}:\\quad\\:$$Vertical$$:\\:x=5,\\:$$Horizontal$$:\\:y=-3$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$\\frac{9-3x}{x-5}:{\\quad}x=5$$", "input": "\\frac{9-3x}{x-5}", "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:$${\\quad}x=5$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{9-3x}{x-5}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x-5=0:{\\quad}x=5$$", "input": "x-5=0", "steps": [ { "type": "interim", "title": "Move $$5\\:$$to the right side", "input": "x-5=0", "result": "x=5", "steps": [ { "type": "step", "primary": "Add $$5$$ to both sides", "result": "x-5+5=0+5" }, { "type": "step", "primary": "Simplify", "result": "x=5" } ], "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=5" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "The vertical asymptotes are:", "result": "x=5" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5dVnUL0bpHaJzHH2MFgu3oeLdNO7q/0LMnCACxnhzN2o38LfSxJ+0AgVLpCSnLX0iSqJ4tnbhgEaZd/MAQce+B+N9uXw2vFVlvX/PjyJMtZMM3iC1ikcCGWL8fa/Z1p5dNg==" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$\\frac{9-3x}{x-5}:{\\quad}y=-3$$", "input": "\\frac{9-3x}{x-5}", "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$$=-3,\\:$$Denominator's leading coefficient$$=1$$", "result": "y=\\frac{-3}{1}" }, { "type": "step", "primary": "The horizontal asymptote is:", "result": "y=-3" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0PihooH8V6rf2M5+iqpeDLe57MsOrW6JJ1Fx1RaqGDaCRzQR72wZm7kDUxdE6YSmfEbr2tjD8GZwje+anH6upXbCvJ91TiauU81vk6/S/bst8ayIFckKrUUyvViQo1lcpJ/Ez7CI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Slant Asymptotes of $$\\frac{9-3x}{x-5}:{\\quad}$$None", "input": "\\frac{9-3x}{x-5}", "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/tmv96KPb2deVtPyMFjlxEz9fqMhq393r2JlY/8nJfmkou6YzfTeQKHeh69S6dnv9vSoUoFB5YPjGJXCZ4z4PoEDjycgk6/FVZFHXHld/q/KN0reRuhNQL6cWsef56JRciAgKzOQ==" } }, { "type": "step", "result": "\\mathrm{Vertical}:\\:x=5,\\:\\mathrm{Horizontal}:\\:y=-3" } ], "meta": { "solvingClass": "Function Asymptotes" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{9-3x}{x-5}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }