inverse of f(x)=4(x-5)^3

{ "query": { "display": "inverse $$f\\left(x\\right)=4\\left(x-5\\right)^{3}$$", "symbolab_question": "FUNCTION#inverse f(x)=4(x-5)^{3}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\sqrt[3]{\\frac{x}{4}}+5", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$4\\left(x-5\\right)^{3}:{\\quad}\\sqrt[3]{\\frac{x}{4}}+5$$", "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=4\\left(x-5\\right)^{3}" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=4\\left(x-5\\right)^{3}", "result": "x=4\\left(y-5\\right)^{3}", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=4\\left(y-5\\right)^{3}" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZXYXq3oX5+wVa72U6N7Iv6OJDNT2wCif15w2ce06so24ADUha1tXi8qEmnvLTUBI9T/L0MoYg+CUn6oyL3EO7YpV5U35Ldoq28O/2Mp19YBuczb172HvovdBpNcutsB6xUlwt5S0mvTtHzYltVBIUs0=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=4\\left(y-5\\right)^{3}$$", "input": "x=4\\left(y-5\\right)^{3}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "4\\left(y-5\\right)^{3}=x" }, { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4\\left(y-5\\right)^{3}=x", "result": "\\left(y-5\\right)^{3}=\\frac{x}{4}", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4\\left(y-5\\right)^{3}}{4}=\\frac{x}{4}" }, { "type": "step", "primary": "Simplify", "result": "\\left(y-5\\right)^{3}=\\frac{x}{4}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "For $$\\left(g\\left(x\\right)\\right)^n=f\\left(a\\right)$$, n is odd, the solution is $$g\\left(x\\right)=\\sqrt[n]{f\\left(a\\right)}$$" }, { "type": "interim", "title": "Solve $$y-5=\\sqrt[3]{\\frac{x}{4}}:{\\quad}y=\\sqrt[3]{\\frac{x}{4}}+5$$", "input": "y-5=\\sqrt[3]{\\frac{x}{4}}", "steps": [ { "type": "interim", "title": "Move $$5\\:$$to the right side", "input": "y-5=\\sqrt[3]{\\frac{x}{4}}", "result": "y=\\sqrt[3]{\\frac{x}{4}}+5", "steps": [ { "type": "step", "primary": "Add $$5$$ to both sides", "result": "y-5+5=\\sqrt[3]{\\frac{x}{4}}+5" }, { "type": "step", "primary": "Simplify", "result": "y=\\sqrt[3]{\\frac{x}{4}}+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 solution is" }, { "type": "step", "result": "y=\\sqrt[3]{\\frac{x}{4}}+5" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "\\sqrt[3]{\\frac{x}{4}}+5" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "4(x-5)^{3}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }