inverse of x/(x-4)

{ "query": { "display": "inverse $$\\frac{x}{x-4}$$", "symbolab_question": "FUNCTION#inverse \\frac{x}{x-4}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\frac{4x}{x-1}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$\\frac{x}{x-4}:{\\quad}\\frac{4x}{x-1}$$", "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{x}{x-4}" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=\\frac{x}{x-4}", "result": "x=\\frac{y}{y-4}", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=\\frac{y}{y-4}" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZWmm4rDwAIZ66rfF+Men0KvFTftC/LvyeY3gSXVru9zxuT7NM3IgGSg7KvASoIT1+CGdMQWj7euUj19HIJMZ9JN6PZq4pYr+MqS+U8UJjX531Kuuv9CNQvrdvkZ5dLi5r5KKTpUQqDZ50xP6ow8quMY=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=\\frac{y}{y-4}$$", "input": "x=\\frac{y}{y-4}", "steps": [ { "type": "interim", "title": "Multiply both sides by $$y-4$$", "input": "x=\\frac{y}{y-4}", "result": "x\\left(y-4\\right)=y", "steps": [ { "type": "step", "primary": "Multiply both sides by $$y-4$$", "result": "x\\left(y-4\\right)=\\frac{y}{y-4}\\left(y-4\\right)" }, { "type": "step", "primary": "Simplify", "result": "x\\left(y-4\\right)=y" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Expand $$x\\left(y-4\\right):{\\quad}xy-4x$$", "input": "x\\left(y-4\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=x,\\:b=y,\\:c=4$$" ], "result": "=xy-x\\cdot\\:4", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=xy-4x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DMyJLXenj7PP/SPZo77kvAsBmUOGgB0VAXoAXik9qC9EoApuujwRq/Ms3wa4gNT89ZVSO0B13oSpK+zJfqv7kB7MWKURN+43KCOzRc+RXaGM4aEyvdeVSKwtdrA/qVqC" } }, { "type": "step", "result": "xy-4x=y" }, { "type": "interim", "title": "Move $$4x\\:$$to the right side", "input": "xy-4x=y", "result": "xy=y+4x", "steps": [ { "type": "step", "primary": "Add $$4x$$ to both sides", "result": "xy-4x+4x=y+4x" }, { "type": "step", "primary": "Simplify", "result": "xy=y+4x" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$y\\:$$to the left side", "input": "xy=y+4x", "result": "xy-y=4x", "steps": [ { "type": "step", "primary": "Subtract $$y$$ from both sides", "result": "xy-y=y+4x-y" }, { "type": "step", "primary": "Simplify", "result": "xy-y=4x" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72W/Uxg35cBPmqbhazAwD2RC1Hih0bIV8/5v07jR9hMwRFAdePdvMBOsk+NlsRzNZiihwSxlE9ITGEL4YEBT3xrN+7Xi9PtBkCIRqG2WMxzdbwVz5T7/Dyunz8G+H2e0kUsD+pIkOCIZyIU4q8atffPpdu80I8/0M2UIpmdiK+fyijfB5FWyAm04vkUoKGvt9jTM0bCosc64b7oYaAGmQkblnMoJbj9GDzdpS7kuziPWCml5kUDT3mdNbphr0HBlc4IMIfYNwM2JqUy6/LsGfCeT5Rj0kETm/9hJnqz4ZzPOdiqcAK81Ko4/KpyFPiCZ8dptnCLknK03/W5+ZCJKKaQYyBJL2j075R6FHVrdPUhNadgwzuY6Jb0cogSIeKVquUcbinK14Gon5f3QBcn/JKo0TG2uNuGbUn+MgmRK5xMRvkr802rxe23zDsqRyozsU5EEd95TCm58cCS2zmqfAHw==" } }, { "type": "interim", "title": "Factor $$xy-y:{\\quad}y\\left(x-1\\right)$$", "input": "xy-y", "steps": [ { "type": "step", "primary": "Factor out common term $$y$$", "result": "=y\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "y\\left(x-1\\right)=4x" }, { "type": "interim", "title": "Divide both sides by $$x-1$$", "input": "y\\left(x-1\\right)=4x", "result": "y=\\frac{4x}{x-1}", "steps": [ { "type": "step", "primary": "Divide both sides by $$x-1$$", "result": "\\frac{y\\left(x-1\\right)}{x-1}=\\frac{4x}{x-1}" }, { "type": "step", "primary": "Simplify", "result": "y=\\frac{4x}{x-1}" } ], "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{4x}{x-1}" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{x}{x-4}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }