asymptotes of f(x)=(x^2-x)/(x^2-8x+7)

{ "query": { "display": "asymptotes $$f\\left(x\\right)=\\frac{x^{2}-x}{x^{2}-8x+7}$$", "symbolab_question": "FUNCTION#asymptotes f(x)=\\frac{x^{2}-x}{x^{2}-8x+7}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "asymptotes", "default": "\\mathrm{Vertical}: x=7,\\mathrm{Horizontal}: y=1", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Asymptotes of $$\\frac{x^{2}-x}{x^{2}-8x+7}:\\quad\\:$$Vertical$$:\\:x=7,\\:$$Horizontal$$:\\:y=1$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$\\frac{x^{2}-x}{x^{2}-8x+7}:{\\quad}x=7$$", "input": "\\frac{x^{2}-x}{x^{2}-8x+7}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{x^{2}-x}{x^{2}-8x+7}:{\\quad}\\frac{x}{x-7}$$", "input": "\\frac{x^{2}-x}{x^{2}-8x+7}", "steps": [ { "type": "interim", "title": "Factor out common term $$x:{\\quad}x\\left(x-1\\right)$$", "input": "x^{2}-x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-1\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=\\frac{x\\left(x-1\\right)}{x^{2}-8x+7}" }, { "type": "interim", "title": "Factor $$x^{2}-8x+7:{\\quad}\\left(x-1\\right)\\left(x-7\\right)$$", "input": "x^{2}-8x+7", "result": "=\\frac{x\\left(x-1\\right)}{\\left(x-1\\right)\\left(x-7\\right)}", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}-8x+7", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=1,\\:b=-8,\\:c=7$$", "$$u*v=7,\\:u+v=-8$$" ] }, { "type": "interim", "title": "Factors of $$7:{\\quad}1,\\:7$$", "input": "7", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$7:{\\quad}7$$", "input": "7", "steps": [ { "type": "step", "primary": "$$7$$ is a prime number, therefore no factorization is possible", "result": "=7" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsuo5jZiW0T+uPgKywRVRozwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJq+H1JIem2SYFK9CRPpgbIy/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$7$$", "result": "1,\\:7" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$7:{\\quad}-1,\\:-7$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-7" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=7,\\:$$check if $$u+v=-8$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=7:\\quad\\:u*v=7,\\:u+v=8\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=-1,\\:v=-7:\\quad\\:u*v=7,\\:u+v=-8\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-1,\\:v=-7" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}-x\\right)+\\left(-7x+7\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwsNpe/XKPNQORsrtQ9IQwx4sjvX7KVUO/AeCFSId4S337bTjH0JhPfEDoyROh2MmxOR81aIMFSjUznRIvxWbLKFjgo1uKy4NXD9iTXkwx6NR/qkG+L3juhyxymvyBMr8LGWTzraAqG8V/6qnGWqib34sFBI3wQIQWRxYDauqaCe0rVZRNf6poD19Z1jpJbHesg==" } }, { "type": "step", "result": "=\\left(x^{2}-x\\right)+\\left(-7x+7\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}-x:\\quad\\:x\\left(x-1\\right)$$", "input": "x^{2}-x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-7\\:$$from $$-7x+7:\\quad\\:-7\\left(x-1\\right)$$", "input": "-7x+7", "steps": [ { "type": "step", "primary": "Factor out common term $$-7$$", "result": "=-7\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jLClz87UseCnMWy8o88tsZN1pXT08zEQpn0WJ6CFMXD9pNnvGSE4c+RSRHJvBJPwbpAQMmevRAGPdN6Y9PB3FCGk8vIJisuT2N3pfkW1JpawooBe7OKH4/NA59T9vDux8GdXw+O6EU4O4Mh/TBEcfKJAGfeDtxrLItV85e3SD00=" } }, { "type": "step", "result": "=x\\left(x-1\\right)-7\\left(x-1\\right)" }, { "type": "step", "primary": "Factor out common term $$x-1$$", "result": "=\\left(x-1\\right)\\left(x-7\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$x-1$$", "result": "=\\frac{x}{x-7}" } ], "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:$${\\quad}x=7$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{x}{x-7}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x-7=0:{\\quad}x=7$$", "input": "x-7=0", "steps": [ { "type": "interim", "title": "Move $$7\\:$$to the right side", "input": "x-7=0", "result": "x=7", "steps": [ { "type": "step", "primary": "Add $$7$$ to both sides", "result": "x-7+7=0+7" }, { "type": "step", "primary": "Simplify", "result": "x=7" } ], "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=7" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "The vertical asymptotes are:", "result": "x=7" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5dVnUL0bpHaJzHH2MFgu3oeJw3WFk6CLT4Tgpo6RWwHlOc/mJRUSGXjIPbk+INESthwJ5wwhCrpBMlDJOmV/ZqpsfvkzNq6HpGXH3yEXiCOJ5Uvn/K0adIu922CUSTnadgoXE4dAv/WReq/Bj3pH37wiwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$\\frac{x^{2}-x}{x^{2}-8x+7}:{\\quad}y=1$$", "input": "\\frac{x^{2}-x}{x^{2}-8x+7}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{x^{2}-x}{x^{2}-8x+7}:{\\quad}\\frac{x}{x-7}$$", "input": "\\frac{x^{2}-x}{x^{2}-8x+7}", "steps": [ { "type": "interim", "title": "Factor out common term $$x:{\\quad}x\\left(x-1\\right)$$", "input": "x^{2}-x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-1\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=\\frac{x\\left(x-1\\right)}{x^{2}-8x+7}" }, { "type": "interim", "title": "Factor $$x^{2}-8x+7:{\\quad}\\left(x-1\\right)\\left(x-7\\right)$$", "input": "x^{2}-8x+7", "result": "=\\frac{x\\left(x-1\\right)}{\\left(x-1\\right)\\left(x-7\\right)}", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}-8x+7", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=1,\\:b=-8,\\:c=7$$", "$$u*v=7,\\:u+v=-8$$" ] }, { "type": "interim", "title": "Factors of $$7:{\\quad}1,\\:7$$", "input": "7", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$7:{\\quad}7$$", "input": "7", "steps": [ { "type": "step", "primary": "$$7$$ is a prime number, therefore no factorization is possible", "result": "=7" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsuo5jZiW0T+uPgKywRVRozwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJq+H1JIem2SYFK9CRPpgbIy/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$7$$", "result": "1,\\:7" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$7:{\\quad}-1,\\:-7$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-7" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=7,\\:$$check if $$u+v=-8$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=7:\\quad\\:u*v=7,\\:u+v=8\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=-1,\\:v=-7:\\quad\\:u*v=7,\\:u+v=-8\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-1,\\:v=-7" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}-x\\right)+\\left(-7x+7\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwsNpe/XKPNQORsrtQ9IQwx4sjvX7KVUO/AeCFSId4S337bTjH0JhPfEDoyROh2MmxOR81aIMFSjUznRIvxWbLKFjgo1uKy4NXD9iTXkwx6NR/qkG+L3juhyxymvyBMr8LGWTzraAqG8V/6qnGWqib34sFBI3wQIQWRxYDauqaCe0rVZRNf6poD19Z1jpJbHesg==" } }, { "type": "step", "result": "=\\left(x^{2}-x\\right)+\\left(-7x+7\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}-x:\\quad\\:x\\left(x-1\\right)$$", "input": "x^{2}-x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-7\\:$$from $$-7x+7:\\quad\\:-7\\left(x-1\\right)$$", "input": "-7x+7", "steps": [ { "type": "step", "primary": "Factor out common term $$-7$$", "result": "=-7\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jLClz87UseCnMWy8o88tsZN1pXT08zEQpn0WJ6CFMXD9pNnvGSE4c+RSRHJvBJPwbpAQMmevRAGPdN6Y9PB3FCGk8vIJisuT2N3pfkW1JpawooBe7OKH4/NA59T9vDux8GdXw+O6EU4O4Mh/TBEcfKJAGfeDtxrLItV85e3SD00=" } }, { "type": "step", "result": "=x\\left(x-1\\right)-7\\left(x-1\\right)" }, { "type": "step", "primary": "Factor out common term $$x-1$$", "result": "=\\left(x-1\\right)\\left(x-7\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$x-1$$", "result": "=\\frac{x}{x-7}" } ], "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$$=1.\\:$$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$$=1,\\:$$Denominator's leading coefficient$$=1$$", "result": "y=\\frac{1}{1}" }, { "type": "step", "primary": "The horizontal asymptote is:", "result": "y=1" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0PihooH8V6rf2M5+iqpeDLe57MtJ5irzRFsW7GUGq2eovJEEQRWv5fZbS/IgRhkOWuSsrYT3IXPNzJorOYYEGzSCK+VBNmPozZ0OWvMjCTqJf2s1wbhivzD47UeI6IIfI4ahyFIiDinipXSQj6SDMsogttUEYQzMKX5hcIwf/7h4h4MF" } }, { "type": "interim", "title": "Slant Asymptotes of $$\\frac{x^{2}-x}{x^{2}-8x+7}:{\\quad}$$None", "input": "\\frac{x^{2}-x}{x^{2}-8x+7}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{x^{2}-x}{x^{2}-8x+7}:{\\quad}\\frac{x}{x-7}$$", "input": "\\frac{x^{2}-x}{x^{2}-8x+7}", "steps": [ { "type": "interim", "title": "Factor out common term $$x:{\\quad}x\\left(x-1\\right)$$", "input": "x^{2}-x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-1\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=\\frac{x\\left(x-1\\right)}{x^{2}-8x+7}" }, { "type": "interim", "title": "Factor $$x^{2}-8x+7:{\\quad}\\left(x-1\\right)\\left(x-7\\right)$$", "input": "x^{2}-8x+7", "result": "=\\frac{x\\left(x-1\\right)}{\\left(x-1\\right)\\left(x-7\\right)}", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}-8x+7", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=1,\\:b=-8,\\:c=7$$", "$$u*v=7,\\:u+v=-8$$" ] }, { "type": "interim", "title": "Factors of $$7:{\\quad}1,\\:7$$", "input": "7", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$7:{\\quad}7$$", "input": "7", "steps": [ { "type": "step", "primary": "$$7$$ is a prime number, therefore no factorization is possible", "result": "=7" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsuo5jZiW0T+uPgKywRVRozwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJq+H1JIem2SYFK9CRPpgbIy/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$7$$", "result": "1,\\:7" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$7:{\\quad}-1,\\:-7$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-7" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=7,\\:$$check if $$u+v=-8$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=7:\\quad\\:u*v=7,\\:u+v=8\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=-1,\\:v=-7:\\quad\\:u*v=7,\\:u+v=-8\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-1,\\:v=-7" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}-x\\right)+\\left(-7x+7\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwsNpe/XKPNQORsrtQ9IQwx4sjvX7KVUO/AeCFSId4S337bTjH0JhPfEDoyROh2MmxOR81aIMFSjUznRIvxWbLKFjgo1uKy4NXD9iTXkwx6NR/qkG+L3juhyxymvyBMr8LGWTzraAqG8V/6qnGWqib34sFBI3wQIQWRxYDauqaCe0rVZRNf6poD19Z1jpJbHesg==" } }, { "type": "step", "result": "=\\left(x^{2}-x\\right)+\\left(-7x+7\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}-x:\\quad\\:x\\left(x-1\\right)$$", "input": "x^{2}-x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-7\\:$$from $$-7x+7:\\quad\\:-7\\left(x-1\\right)$$", "input": "-7x+7", "steps": [ { "type": "step", "primary": "Factor out common term $$-7$$", "result": "=-7\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jLClz87UseCnMWy8o88tsZN1pXT08zEQpn0WJ6CFMXD9pNnvGSE4c+RSRHJvBJPwbpAQMmevRAGPdN6Y9PB3FCGk8vIJisuT2N3pfkW1JpawooBe7OKH4/NA59T9vDux8GdXw+O6EU4O4Mh/TBEcfKJAGfeDtxrLItV85e3SD00=" } }, { "type": "step", "result": "=x\\left(x-1\\right)-7\\left(x-1\\right)" }, { "type": "step", "primary": "Factor out common term $$x-1$$", "result": "=\\left(x-1\\right)\\left(x-7\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$x-1$$", "result": "=\\frac{x}{x-7}" } ], "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$$=1.\\:$$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/tmv96KL5mikwnGs7EvIxpFPRJ09vKl0sou07ssJ5yiSYDqsaRFiiplRl01eNDr25Q1kVbGERYs2eAmakdgiiTqjlqN2x1TiauU81vk6/S/bst8ayIehbQllT6EVTmn8ENBm8dCPfNypfTECYzHBBlYe2piNk=" } }, { "type": "step", "result": "\\mathrm{Vertical}:\\:x=7,\\:\\mathrm{Horizontal}:\\:y=1" } ], "meta": { "solvingClass": "Function Asymptotes" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{x^{2}-x}{x^{2}-8x+7}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }