inverse of f(x)=(e^{4x})/(3+e^{4x)}

{ "query": { "display": "inverse $$f\\left(x\\right)=\\frac{e^{4x}}{3+e^{4x}}$$", "symbolab_question": "FUNCTION#inverse f(x)=\\frac{e^{4x}}{3+e^{4x}}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\frac{\\ln(-\\frac{3x}{x-1})}{4}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$\\frac{e^{4x}}{3+e^{4x}}:{\\quad}\\frac{\\ln\\left(-\\frac{3x}{x-1}\\right)}{4}$$", "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{e^{4x}}{3+e^{4x}}" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=\\frac{e^{4x}}{3+e^{4x}}", "result": "x=\\frac{e^{4y}}{3+e^{4y}}", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=\\frac{e^{4y}}{3+e^{4y}}" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZaC2Sj722K0uj+dIwLxhH/OCmlR7phRqgT9ap5OYXkP0CHXJZWsJ6tajRPnKMCZK0waLbW7tmrou0wOIs8jN2rWjeh7+jKEzLb7VNCEMF3Z/HExhGNbuMueeg7Wus8Us4algRF/EcMQGCXb0XYsMFo7myoqA4TOJ4PpM8lDltkgl" } }, { "type": "interim", "title": "Solve for $$y,\\:x=\\frac{e^{4y}}{3+e^{4y}}$$", "input": "x=\\frac{e^{4y}}{3+e^{4y}}", "steps": [ { "type": "step", "primary": "Multiply both sides by $$3+e^{4y}$$", "result": "x\\left(3+e^{4y}\\right)=\\frac{e^{4y}}{3+e^{4y}}\\left(3+e^{4y}\\right)" }, { "type": "step", "primary": "Simplify", "result": "x\\left(3+e^{4y}\\right)=e^{4y}" }, { "type": "interim", "title": "Expand $$x\\left(3+e^{4y}\\right):{\\quad}3x+xe^{4y}$$", "input": "x\\left(3+e^{4y}\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=x,\\:b=3,\\:c=e^{4y}$$" ], "result": "=x\\cdot\\:3+xe^{4y}", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=3x+xe^{4y}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TdOCZr7CxA7b7s/37xHS0M+G4ofi6jCZ7MTGotGZAbneRv0Qg6eY1kN4TiDkMoxs+yxfSSpJ0xijNZCv3R7nSdbA+zX4bD3u3gx65o2NJhPilrUgNjkPCHSpqpW0mrsecnuScSt7xR1aTqoTwZUyVw==" } }, { "type": "step", "result": "3x+xe^{4y}=e^{4y}" }, { "type": "interim", "title": "Move $$3x\\:$$to the right side", "input": "3x+xe^{4y}=e^{4y}", "result": "xe^{4y}=e^{4y}-3x", "steps": [ { "type": "step", "primary": "Subtract $$3x$$ from both sides", "result": "3x+xe^{4y}-3x=e^{4y}-3x" }, { "type": "step", "primary": "Simplify", "result": "xe^{4y}=e^{4y}-3x" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$e^{4y}\\:$$to the left side", "input": "xe^{4y}=e^{4y}-3x", "result": "xe^{4y}-e^{4y}=-3x", "steps": [ { "type": "step", "primary": "Subtract $$e^{4y}$$ from both sides", "result": "xe^{4y}-e^{4y}=e^{4y}-3x-e^{4y}" }, { "type": "step", "primary": "Simplify", "result": "xe^{4y}-e^{4y}=-3x" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Factor $$xe^{4y}-e^{4y}:{\\quad}e^{4y}\\left(x-1\\right)$$", "input": "xe^{4y}-e^{4y}", "steps": [ { "type": "step", "primary": "Factor out common term $$e^{4y}$$", "result": "=e^{4y}\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "e^{4y}\\left(x-1\\right)=-3x" }, { "type": "interim", "title": "Divide both sides by $$x-1$$", "input": "e^{4y}\\left(x-1\\right)=-3x", "result": "e^{4y}=-\\frac{3x}{x-1}", "steps": [ { "type": "step", "primary": "Divide both sides by $$x-1$$", "result": "\\frac{e^{4y}\\left(x-1\\right)}{x-1}=\\frac{-3x}{x-1}" }, { "type": "step", "primary": "Simplify", "result": "e^{4y}=-\\frac{3x}{x-1}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply exponent rules", "input": "e^{4y}=-\\frac{3x}{x-1}", "result": "4y=\\ln\\left(-\\frac{3x}{x-1}\\right)", "steps": [ { "type": "step", "primary": "If $$f\\left(x\\right)=g\\left(x\\right)$$, then $$\\ln\\left(f\\left(x\\right)\\right)=\\ln\\left(g\\left(x\\right)\\right)$$", "result": "\\ln\\left(e^{4y}\\right)=\\ln\\left(-\\frac{3x}{x-1}\\right)" }, { "type": "step", "primary": "Apply log rule: $$\\ln\\left(e^a\\right)=a$$", "secondary": [ "$$\\ln\\left(e^{4y}\\right)=4y$$" ], "result": "4y=\\ln\\left(-\\frac{3x}{x-1}\\right)", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Apply Exp Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7L/wjoti6Q8HXQVdMd8S223kwN5DcxD5ZUeYVZLmoSy02+deurbgprpiI9/et+AgSoR3idcQYt9tS42OL7FwvXptxQIj88VctLpqHYQzRHTJ4vygjNxDFCNG/XZPtpA1dveIZ5RY4+60ARUG7QWJKa2RLd2VwIqlBNByF6663syTkbx0w2SNK21KuZkvjbddhCRNEHBHYhcH1tZIB5929zA==" } }, { "type": "interim", "title": "Solve $$4y=\\ln\\left(-\\frac{3x}{x-1}\\right):{\\quad}y=\\frac{\\ln\\left(-\\frac{3x}{x-1}\\right)}{4}$$", "input": "4y=\\ln\\left(-\\frac{3x}{x-1}\\right)", "steps": [ { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4y=\\ln\\left(-\\frac{3x}{x-1}\\right)", "result": "y=\\frac{\\ln\\left(-\\frac{3x}{x-1}\\right)}{4}", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4y}{4}=\\frac{\\ln\\left(-\\frac{3x}{x-1}\\right)}{4}" }, { "type": "step", "primary": "Simplify", "result": "y=\\frac{\\ln\\left(-\\frac{3x}{x-1}\\right)}{4}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "y=\\frac{\\ln\\left(-\\frac{3x}{x-1}\\right)}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "\\frac{\\ln\\left(-\\frac{3x}{x-1}\\right)}{4}" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{e^{4x}}{3+e^{4x}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }