inverse of f(x)=-\sqrt[3]{x+1}-5

{ "query": { "display": "inverse $$f\\left(x\\right)=-\\sqrt[3]{x+1}-5$$", "symbolab_question": "FUNCTION#inverse f(x)=-\\sqrt[3]{x+1}-5" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "-x^{3}-15x^{2}-75x-126", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$-\\sqrt[3]{x+1}-5:{\\quad}-x^{3}-15x^{2}-75x-126$$", "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=-\\sqrt[3]{x+1}-5" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=-\\sqrt[3]{x+1}-5", "result": "x=-\\sqrt[3]{y+1}-5", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=-\\sqrt[3]{y+1}-5" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZayaTOEcaTt1WPSU63P0nf+qLYvB9InEtlF80xnGEV+rGQmYVIcZpIrwEyqT9Afp8/wjxGUyXxWBxRqXMq0bSOpN5Aod6Hr1Lp2e/29KhSgUmjqleygtlSszoezKDjz2k7GJwpZAgCxm4WPlcU4xj5k=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=-\\sqrt[3]{y+1}-5$$", "input": "x=-\\sqrt[3]{y+1}-5", "steps": [ { "type": "interim", "title": "Move $$\\sqrt[3]{y+1}\\:$$to the left side", "input": "x=-\\sqrt[3]{y+1}-5", "result": "x+\\sqrt[3]{y+1}=-5", "steps": [ { "type": "step", "primary": "Add $$\\sqrt[3]{y+1}$$ to both sides", "result": "x+\\sqrt[3]{y+1}=-\\sqrt[3]{y+1}-5+\\sqrt[3]{y+1}" }, { "type": "step", "primary": "Simplify", "result": "x+\\sqrt[3]{y+1}=-5" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$x\\:$$to the right side", "input": "x+\\sqrt[3]{y+1}=-5", "result": "\\sqrt[3]{y+1}=-5-x", "steps": [ { "type": "step", "primary": "Subtract $$x$$ from both sides", "result": "x+\\sqrt[3]{y+1}-x=-5-x" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt[3]{y+1}=-5-x" } ], "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 $$3:{\\quad}y+1=-125-75x-15x^{2}-x^{3}$$", "input": "\\sqrt[3]{y+1}=-5-x", "result": "y+1=-125-75x-15x^{2}-x^{3}", "steps": [ { "type": "step", "result": "\\left(\\sqrt[3]{y+1}\\right)^{3}=\\left(-5-x\\right)^{3}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt[3]{y+1}\\right)^{3}:{\\quad}y+1$$", "input": "\\left(\\sqrt[3]{y+1}\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "result": "=\\left(\\left(y+1\\right)^{\\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": "=\\left(y+1\\right)^{\\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+1", "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+eNAid7ACAFUgQhF9g4NcaMWBPfaec1RZIwlCMvbTN26PYgr6ZvcSHthJ+x3ZgtR9gxl5lh0eio4FTI3JXph8YFx3d2CcvL2jSzw1VlRcGX4ka6qIsN7kMj5Tdjw04TGIQOo3p/riQwWJOmWVPULBlvcePk=" } }, { "type": "interim", "title": "Expand $$\\left(-5-x\\right)^{3}:{\\quad}-125-75x-15x^{2}-x^{3}$$", "input": "\\left(-5-x\\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": [ "$$a=-5,\\:\\:b=x$$" ] }, { "type": "step", "result": "=\\left(-5\\right)^{3}-3\\left(-5\\right)^{2}x+3\\left(-5\\right)x^{2}-x^{3}" }, { "type": "interim", "title": "Simplify $$\\left(-5\\right)^{3}-3\\left(-5\\right)^{2}x+3\\left(-5\\right)x^{2}-x^{3}:{\\quad}-125-75x-15x^{2}-x^{3}$$", "input": "\\left(-5\\right)^{3}-3\\left(-5\\right)^{2}x+3\\left(-5\\right)x^{2}-x^{3}", "result": "=-125-75x-15x^{2}-x^{3}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\left(-5\\right)^{3}-3\\left(-5\\right)^{2}x-3\\cdot\\:5x^{2}-x^{3}" }, { "type": "interim", "title": "$$\\left(-5\\right)^{3}=-125$$", "input": "\\left(-5\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=-a^{n},\\:$$if $$n$$ is odd", "secondary": [ "$$\\left(-5\\right)^{3}=-5^{3}$$" ], "result": "=-5^{3}" }, { "type": "step", "primary": "$$5^{3}=125$$", "result": "=-125" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7M+s4PtCxnmWNOd1qLHIriM0ag8T1MwTer44+aCS/ZFDh0JqKr2NAGBq0NMG1WdkAasAVIxxkpEvMOnZ2+yRgnG99K3hL/+8T0L19mwvRv5M=" } }, { "type": "step", "result": "=-125-\\left(-5\\right)^{2}\\cdot\\:3x-3\\cdot\\:5x^{2}-x^{3}" }, { "type": "interim", "title": "$$3\\left(-5\\right)^{2}x=75x$$", "input": "3\\left(-5\\right)^{2}x", "steps": [ { "type": "interim", "title": "$$\\left(-5\\right)^{2}=5^{2}$$", "input": "\\left(-5\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-5\\right)^{2}=5^{2}$$" ], "result": "=5^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wnXrRJl3NfrdytdqTU5D9M0ag8T1MwTer44+aCS/ZFDbaBNHsH8tVTRhOBMFn40PP3LKgyuLDbbzWOwJ+ohiYyajhqXKalkLnehZ6EQ1gMs=" } }, { "type": "step", "result": "=5^{2}\\cdot\\:3x" }, { "type": "step", "primary": "Refine", "result": "=75x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7g45+PvMkbTtyc2A9mu1nRCAn9lkDfZkicUGkO3EF+IpJ0ePks5veAbcqibJK4UqX4Y30EwP2bNRZNqzHIF1vaZ1Dp0A3gYhaztQvwTUjJIE=" } }, { "type": "interim", "title": "$$3\\cdot\\:5x^{2}=15x^{2}$$", "input": "3\\cdot\\:5x^{2}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:5=15$$", "result": "=15x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Pqa9g7iFWCsPekPrS69APgCWKUbvV6WK3fDUgFtg3Q9ckM7vE9CGyXWXs2vbNAIlh0R4kHw1LArZyd/YFzVYTf3wX3yn+wdLlHJ0keLxsU5t9vBrNtg5d3ifY05au68f" } }, { "type": "step", "result": "=-125-75x-15x^{2}-x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77s8qrYL7UC7ogtrWyAHwbjRDCc4ea9KdZf+RpctY+TF1DFKHA6Ddb5RE0Ay7mw9rpVjCuiAuXb1FPodh3Y0nfXql8XXPq6bNQlMm+36iNhmNhq/+SVLn19nsVrYk/wGuEk7YwRD0hobDQfAqSn6vxw==" } }, { "type": "step", "result": "y+1=-125-75x-15x^{2}-x^{3}" } ], "meta": { "interimType": "Take Both Sides To Power Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDaZCNCMlwFw9Zrpnfgo8QWhC+JpftNCBmzyj5vu29G2bj1sE1rUYarwF0JCJubFAmM+9H6w4nD4o2bAjpkSrNz35Kfd9pSoiCMV3FyjoQOXsRJK+roN7IvzTT1lB41ZNnpAiWE2SAAq0xBgJlW3D+fE6rKyX5rIBA3+V2dJrcPDGqJUOoNMB669Mch1Aixn/NGdVDSSLPbiYV+Bav9GDaQvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "Solve $$y+1=-125-75x-15x^{2}-x^{3}:{\\quad}y=-x^{3}-15x^{2}-75x-126$$", "input": "y+1=-125-75x-15x^{2}-x^{3}", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "y+1=-125-75x-15x^{2}-x^{3}", "result": "y=-x^{3}-15x^{2}-75x-126", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "y+1-1=-125-75x-15x^{2}-x^{3}-1" }, { "type": "step", "primary": "Simplify", "result": "y=-x^{3}-15x^{2}-75x-126" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "y=-x^{3}-15x^{2}-75x-126" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "-x^{3}-15x^{2}-75x-126" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "-\\sqrt[3]{x+1}-5" }, "showViewLarger": true } }, "meta": { "showVerify": true } }