range of f(x)=(8x)/(9x-1)

{ "query": { "display": "range $$f\\left(x\\right)=\\frac{8x}{9x-1}$$", "symbolab_question": "FUNCTION#range f(x)=\\frac{8x}{9x-1}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "range", "default": "f(x)<\\frac{8}{9}\\lor f(x)>\\frac{8}{9}", "interval": "(-\\infty ,\\frac{8}{9})\\cup (\\frac{8}{9},\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Range of $$\\frac{8x}{9x-1}:{\\quad}f\\left(x\\right)<\\frac{8}{9}\\lor\\:f\\left(x\\right)>\\frac{8}{9}$$", "steps": [ { "type": "definition", "title": "Function range definition", "text": "The set of values of the dependent variable for which a function is defined" }, { "type": "step", "primary": "The function range is the combined domain of the inverse functions" }, { "type": "step", "primary": "Find the inverse function(s) of: $$\\frac{8x}{9x-1}$$" }, { "type": "interim", "title": "Inverse of $$\\frac{8x}{9x-1}:{\\quad}\\frac{x}{9x-8}$$", "steps": [ { "type": "definition", "title": "Function Inverse definition", "text": "A function g is the inverse of function f if for $$y=f\\left(x\\right),\\:\\:x=g\\left(y\\right)\\:$$" }, { "type": "step", "result": "y=\\frac{8x}{9x-1}" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=\\frac{8x}{9x-1}", "result": "x=\\frac{8y}{9y-1}", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=\\frac{8y}{9y-1}" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZZZ/AHPUxGsBjBLoBvG3NA6XLy4cXbcOhDbKXjjQr57G3SdWT3E2VyK1nevy294PYnvEZy18Tqg4s9eiaTh1Kha9dDZkXCs5+Cxg1YNvJNPUUC7qGPVSkwQY4sUNa/kEA/selcHNpBzTkjlrYjmh3ko=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=\\frac{8y}{9y-1}$$", "input": "x=\\frac{8y}{9y-1}", "steps": [ { "type": "interim", "title": "Multiply both sides by $$9y-1$$", "input": "x=\\frac{8y}{9y-1}", "result": "x\\left(9y-1\\right)=8y", "steps": [ { "type": "step", "primary": "Multiply both sides by $$9y-1$$", "result": "x\\left(9y-1\\right)=\\frac{8y}{9y-1}\\left(9y-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "x\\left(9y-1\\right)=8y" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Expand $$x\\left(9y-1\\right):{\\quad}9xy-x$$", "input": "x\\left(9y-1\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=x,\\:b=9y,\\:c=1$$" ], "result": "=x\\cdot\\:9y-x\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=9xy-1\\cdot\\:x" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=9xy-x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75e62/U1fKTRee7GPa/r/c3cUJadsLvGcDY7IUPYXjf6zs903yhxK0NInTwR7JvSRyM/29l7SHlwNleTQoRn0qBJyf8zawtgEaDEKWrMLEzZe1haextZIZQmgr7KZ5u9K" } }, { "type": "step", "result": "9xy-x=8y" }, { "type": "interim", "title": "Move $$x\\:$$to the right side", "input": "9xy-x=8y", "result": "9xy=8y+x", "steps": [ { "type": "step", "primary": "Add $$x$$ to both sides", "result": "9xy-x+x=8y+x" }, { "type": "step", "primary": "Simplify", "result": "9xy=8y+x" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73EcCVWwkV3f4SjSGwo0YEjigGhg3/px8WfZYMphU3q4VwAsxHZHO18XPmtAo3dTQ/uanPA/ambaYVekTTf2mJOU23Rk1g6ZN9Paip+AOeIUtkBhB366XZw9I2a8vg8150hYzF6vxjSZoMSInCY/PPQbO95/2yiZOrh8Vdx/OHPtKz7fyzzEcLgWbHTMEh+XGDVF69MzLWvSCxGPuKrwBXMaq4/U7xXc0Y9xEMG9jKdHxzpx7J2OJL1RpHGPs755Ps/m5+sb5WWiKPMccgeiF/qSDgeCk/konC01Eg5DnRCeZ6gaGk8GkScU/NEaFqGM+O/AP8XyBLdl53hknbfBDxXqfNbvVQK6fvzqwGmsGkr6G+xF+RZome+GPni2G5KatP698Ye3ys3A/5HLdOiC1F3JenIILBQgjB0g75U/xyXtk0le+LDpk2ASHygEoSwVuDCL/BDJctkaGfWUvEGRUtcyy8IMOFuT0NAkYbTmvqvjcIzt++MOgBXqkfQaiNHmrsjovWk4zEFwmftGRfxnjOl++sNxoNBikwZw3A5RqS3BkS3dlcCKpQTQcheuut7Mk1SS/jz8vttSq+qoI6yLjTbosG1Pgfp/tzvgiDMWD+jHah5rV1PysnJxsQMCqBsNw" } }, { "type": "interim", "title": "Move $$8y\\:$$to the left side", "input": "9xy=8y+x", "result": "9xy-8y=x", "steps": [ { "type": "step", "primary": "Subtract $$8y$$ from both sides", "result": "9xy-8y=8y+x-8y" }, { "type": "step", "primary": "Simplify", "result": "9xy-8y=x" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Factor $$9xy-8y:{\\quad}y\\left(9x-8\\right)$$", "input": "9xy-8y", "steps": [ { "type": "step", "primary": "Factor out common term $$y$$", "result": "=y\\left(9x-8\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "y\\left(9x-8\\right)=x" }, { "type": "interim", "title": "Divide both sides by $$9x-8$$", "input": "y\\left(9x-8\\right)=x", "result": "y=\\frac{x}{9x-8}", "steps": [ { "type": "step", "primary": "Divide both sides by $$9x-8$$", "result": "\\frac{y\\left(9x-8\\right)}{9x-8}=\\frac{x}{9x-8}" }, { "type": "step", "primary": "Simplify", "result": "y=\\frac{x}{9x-8}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "\\frac{x}{9x-8}" } ], "meta": { "solvingClass": "Function Inverse", "interimType": "Function Inverse Top 1Eq" } }, { "type": "step", "result": "\\frac{x}{9x-8}" }, { "type": "step", "primary": "Find the domain of each inverse function" }, { "type": "interim", "title": "Domain of $$\\frac{x}{9x-8}\\::{\\quad}x<\\frac{8}{9}\\lor\\:x>\\frac{8}{9}$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=\\frac{8}{9}$$", "input": "\\frac{x}{9x-8}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{x}{9x-8}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$9x-8=0:{\\quad}x=\\frac{8}{9}$$", "input": "9x-8=0", "steps": [ { "type": "interim", "title": "Move $$8\\:$$to the right side", "input": "9x-8=0", "result": "9x=8", "steps": [ { "type": "step", "primary": "Add $$8$$ to both sides", "result": "9x-8+8=0+8" }, { "type": "step", "primary": "Simplify", "result": "9x=8" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$9$$", "input": "9x=8", "result": "x=\\frac{8}{9}", "steps": [ { "type": "step", "primary": "Divide both sides by $$9$$", "result": "\\frac{9x}{9}=\\frac{8}{9}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{8}{9}" } ], "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{8}{9}" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6soR7Fp9tE97oF13SXjPKZWD0yqNoTWLwGuqxO0C0cFV5ApqFyraVAYbrSDuxrlIa8J9e1u2YCYn2ZgVDfYZK5CRcW/agFqpLn6h+exKVoarqFKStRvo2EnGegTHxetYg+iyAQ4h/yCDl+R95s7zDj54/wSSw2Uq+t8eJqx4ZKnz6/Pv5yxck5A630rhd7hMZA==" } }, { "type": "step", "primary": "The function domain", "result": "x<\\frac{8}{9}\\lor\\:x>\\frac{8}{9}" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "step", "primary": "Combine the intervals", "result": "f\\left(x\\right)<\\frac{8}{9}\\lor\\:f\\left(x\\right)>\\frac{8}{9}" } ], "meta": { "solvingClass": "Function Range" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{8x}{9x-1}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }