inverse of 16+\sqrt[3]{x}

{ "query": { "display": "inverse $$16+\\sqrt[3]{x}$$", "symbolab_question": "FUNCTION#inverse 16+\\sqrt[3]{x}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "x^{3}-48x^{2}+768x-4096", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$16+\\sqrt[3]{x}:{\\quad}x^{3}-48x^{2}+768x-4096$$", "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=16+\\sqrt[3]{x}" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=16+\\sqrt[3]{x}", "result": "x=16+\\sqrt[3]{y}", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=16+\\sqrt[3]{y}" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZQSFH92uLnxKeEEohRu3ExVResVaS4W1dvfd3dEwTUwpHtCSq0ra2jw5oIEnndJe8QmgbJx0WvCBTjJ6oci78gqrve3E7cDlwD8G9VYfu6duV3mk0kzsCUJ7KXIclF5bXIvTKrluJAbyVSusE7F4pyU=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=16+\\sqrt[3]{y}$$", "input": "x=16+\\sqrt[3]{y}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "16+\\sqrt[3]{y}=x" }, { "type": "interim", "title": "Move $$16\\:$$to the right side", "input": "16+\\sqrt[3]{y}=x", "result": "\\sqrt[3]{y}=x-16", "steps": [ { "type": "step", "primary": "Subtract $$16$$ from both sides", "result": "16+\\sqrt[3]{y}-16=x-16" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt[3]{y}=x-16" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Take both sides of the equation to the power of $$3:{\\quad}y=x^{3}-48x^{2}+768x-4096$$", "input": "\\sqrt[3]{y}=x-16", "result": "y=x^{3}-48x^{2}+768x-4096", "steps": [ { "type": "step", "result": "\\left(\\sqrt[3]{y}\\right)^{3}=\\left(x-16\\right)^{3}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt[3]{y}\\right)^{3}:{\\quad}y$$", "input": "\\left(\\sqrt[3]{y}\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "result": "=\\left(y^{\\frac{1}{3}}\\right)^{3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=y^{\\frac{1}{3}\\cdot\\:3}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{3}\\cdot\\:3=1$$", "input": "\\frac{1}{3}\\cdot\\:3", "result": "=y", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3}{3}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71Hcipo16ybrRdbQSFyXFLyPpFLppSOOOwknMsNOrv36rju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6ndrE5pRJHjGLuD4z7ETvZP4l/Cp3Oq6hZvUc7OAI8rhp" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vz0OwVmz7FbkxdUH+eNAiYi5YrgnoZalaTFi5QFm3kjptRYmj6WUpgaKrNIresW2wNmnTNPfSXaFwWcdWyIh039H/SFX0n2Ge344m0fgESR0U1L0UOfZXYZXOhi7eOgpWUk0APCxstnPXDDD+I99IhJO2MEQ9IaGw0HwKkp+r8c=" } }, { "type": "interim", "title": "Expand $$\\left(x-16\\right)^{3}:{\\quad}x^{3}-48x^{2}+768x-4096$$", "input": "\\left(x-16\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply Perfect Cube Formula: $$\\left(a-b\\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}$$", "secondary": [ "$$a=x,\\:\\:b=16$$" ] }, { "type": "step", "result": "=x^{3}-3x^{2}\\cdot\\:16+3x\\cdot\\:16^{2}-16^{3}" }, { "type": "interim", "title": "Simplify $$x^{3}-3x^{2}\\cdot\\:16+3x\\cdot\\:16^{2}-16^{3}:{\\quad}x^{3}-48x^{2}+768x-4096$$", "input": "x^{3}-3x^{2}\\cdot\\:16+3x\\cdot\\:16^{2}-16^{3}", "result": "=x^{3}-48x^{2}+768x-4096", "steps": [ { "type": "interim", "title": "$$3x^{2}\\cdot\\:16=48x^{2}$$", "input": "3x^{2}\\cdot\\:16", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:16=48$$", "result": "=48x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ap3jqZBQjp6oXNZCipg0wZvB5hGuT5BE+tmykEcm922PYXFoSBeiXHFdzmIy/m85DzRG5i2RnXJK7EZvbhYwA1qV6lCGXUbqgIuYQGu4ucSlwuNjCiBW5VdfdssSaGrS" } }, { "type": "interim", "title": "$$3x\\cdot\\:16^{2}=768x$$", "input": "3x\\cdot\\:16^{2}", "steps": [ { "type": "step", "primary": "$$16^{2}=256$$", "result": "=3\\cdot\\:256x" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:256=768$$", "result": "=768x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76oz9f8lJIhbFMJxf4DbTxnyRHuGw7+tM5METTDj6vVFs7+qbfrxt8rSqamrTQm34/tgVg0HyGLEfBfsrw6s6bOfMAfOiWjIGMswzugv8zV1nM4sZ2GmZwN+2qpNZPYwi" } }, { "type": "interim", "title": "$$16^{3}=4096$$", "input": "16^{3}", "steps": [ { "type": "step", "primary": "$$16^{3}=4096$$", "result": "=4096" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NlzgP4xK6XKE2MnifcZl43WD310L1+P2yDQQfMEhENGZSt/Vg+BYOTrbHPUc3ZGW6Rz/Y/P5SaM9arECIK2LAz6LkrPIjPs72BLkFK0Xjd4=" } }, { "type": "step", "result": "=x^{3}-48x^{2}+768x-4096" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lmQyE7DW8GNUwsqzTAmheDRDCc4ea9KdZf+RpctY+TF1DFKHA6Ddb5RE0Ay7mw9rpVjCuiAuXb1FPodh3Y0nfXql8XXPq6bNQlMm+36iNhlc1SXAMowiJ5IcKRMw0RrjEk7YwRD0hobDQfAqSn6vxw==" } }, { "type": "step", "result": "y=x^{3}-48x^{2}+768x-4096" } ], "meta": { "interimType": "Take Both Sides To Power Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDaZCNCMlwFw9Zrpnfgo8QWnHJ2uUAguAUCYS4J53FIo3XDg1Z3KZYaDri4vsnz807vigVp2xOx9s+jfloW9U/jXokE6dSvIJ//kvHPbEGI3CJptYOj2tkAA513sqP0t86uIBSJTOvEsqoSijjiwl7XHmUxKGNKdnUtrUsE68dU3dswNSHArBpilMGBzF+a9/uQeV+/q4UnElD5dYMfcAfx" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "x^{3}-48x^{2}+768x-4096" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "16+\\sqrt[3]{x}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }