inverse of f(x)=x^2+x+6

{ "query": { "display": "inverse $$f\\left(x\\right)=x^{2}+x+6$$", "symbolab_question": "FUNCTION#inverse f(x)=x^{2}+x+6" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\frac{-1+\\sqrt{4x-23}}{2},\\frac{-1-\\sqrt{4x-23}}{2}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$x^{2}+x+6:{\\quad}\\frac{-1+\\sqrt{4x-23}}{2},\\:\\frac{-1-\\sqrt{4x-23}}{2}$$", "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=x^{2}+x+6" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=x^{2}+x+6", "result": "x=y^{2}+y+6", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=y^{2}+y+6" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZeHHLW12vgXltqRXVpme1YkRbanK99cKyYwa/mBTZldqH9GE5izqXI/bDuw3oLXKo2RLd2VwIqlBNByF6663syTDkDkXF8vCx6Q9CRnz+NFaVBWbEKIE9nJ8YkcdZveNBKw7n5+P/GePwavH269Q9eA=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=y^{2}+y+6$$", "input": "x=y^{2}+y+6", "steps": [ { "type": "step", "primary": "Switch sides", "result": "y^{2}+y+6=x" }, { "type": "interim", "title": "Move $$x\\:$$to the left side", "input": "y^{2}+y+6=x", "result": "y^{2}+y+6-x=0", "steps": [ { "type": "step", "primary": "Subtract $$x$$ from both sides", "result": "y^{2}+y+6-x=x-x" }, { "type": "step", "primary": "Simplify", "result": "y^{2}+y+6-x=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "y^{2}+y+6-x=0", "result": "{y}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\cdot\\:1\\cdot\\:\\left(6-x\\right)}}{2\\cdot\\:1}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=1,\\:b=1,\\:c=6-x$$", "result": "{y}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\cdot\\:1\\cdot\\:\\left(6-x\\right)}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jcj3Xaxo+viXN2ZB0Y3z7mHF536AQ8lmBG6lfLTTFPeo/JI5bBgpgExN510TA5cywo6iOxOQeL70yTMBl67ndVrsdW4QlnVHZaYhaD1DKw8oBMoa4+kIzT8OG9yaVGqW+fpCG9NdgCyp7Phu3UJ7Bf79XCQqC9trbAbyjgFu1hJ21IbBeVbgqnI67Ha54Eath8RwJwqul4ltSrD2yHVs6awe4Ja6NNzJubzLiWy1zcCKK5O8dkw0yJ1ERokKuRYZwpht1PqUmNXJTNPGrd7u2Ma0We1fWFSuw5REfdLmIYmmbGr3f95PRd/y8h3lUmP0Da1kmDzRF8AEdB9AbomzM4NmnreEEaGOyS+4N+t9v1n4zjvBTCh95txzZviGUMMJeB3JNIgAt4okc53Rulv5qSxd/3OQYxmsr5R6AryEWpbNtW+xt7YXAHnnUXfOhpt2FylN12FcHo+l2d9qBsS/3H5z9JkYomU9P1YGTFRFqDdN5Aod6Hr1Lp2e/29KhSgUFdl3XxkGszefj4hqCRIN0mp6GT0oXcCVg7/8De2Bm0CJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Simplify $$\\sqrt{1^{2}-4\\cdot\\:1\\cdot\\:\\left(6-x\\right)}:{\\quad}\\sqrt{4x-23}$$", "input": "\\sqrt{1^{2}-4\\cdot\\:1\\cdot\\:\\left(6-x\\right)}", "result": "{y}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{4x-23}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\sqrt{1-4\\cdot\\:1\\cdot\\:\\left(-x+6\\right)}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=\\sqrt{1-4\\left(-x+6\\right)}" }, { "type": "interim", "title": "Expand $$1-4\\left(6-x\\right):{\\quad}4x-23$$", "input": "1-4\\left(6-x\\right)", "result": "=\\sqrt{4x-23}", "steps": [ { "type": "interim", "title": "Expand $$-4\\left(6-x\\right):{\\quad}-24+4x$$", "input": "-4\\left(6-x\\right)", "result": "=1-24+4x", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=-4,\\:b=6,\\:c=x$$" ], "result": "=-4\\cdot\\:6-\\left(-4\\right)x", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a$$" ], "result": "=-4\\cdot\\:6+4x" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:6=24$$", "result": "=-24+4x" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jMXBypS1THlSEplLlk7UTgOfOVs9mPIqDLV5QIWwt3la/fnxS6mbzCbTRyDc+7J+72wZm7kDUxdE6YSmfEbr2nLrCRinOfNP15EAFDR7RJ5pXnfLWHHOG0/iFeWIlqHp" } }, { "type": "step", "primary": "Subtract the numbers: $$1-24=-23$$", "result": "=4x-23" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vQ1eni2a92ULcdK1/BEk2M0ag8T1MwTer44+aCS/ZFAK6rmTxUiOCZAom7ZYk/mu72wZm7kDUxdE6YSmfEbr2gqBq5oYbkbp0jpsXvPVPcgEht7HGr0skRNke0gtuSVP" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Separate the solutions", "result": "{y}_{1}=\\frac{-1+\\sqrt{4x-23}}{2\\cdot\\:1},\\:{y}_{2}=\\frac{-1-\\sqrt{4x-23}}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$y=\\frac{-1+\\sqrt{4x-23}}{2\\cdot\\:1}:{\\quad}\\frac{-1+\\sqrt{4x-23}}{2}$$", "input": "\\frac{-1+\\sqrt{4x-23}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-1+\\sqrt{4x-23}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gI32wa8JgqOZw2m0j1OS96LtgQ/buQxiZtxG95g5rqa2yYtEqeT7UjI2zAoMd7PXcJChiVhDxT5N/LHSTLMjyPHCJ9oPFiYc8vf3hwgls/0h8LHQeR3zCTVr2pbPKm/fJMdNUVWcJhfOcXhVwo2bw+PwJ+pefpTeaFGZC/bAr3pBPUDwBnNzGwo5/z2vgwUCbGUhD9+gDQg079F600Xu5Q==" } }, { "type": "interim", "title": "$$y=\\frac{-1-\\sqrt{4x-23}}{2\\cdot\\:1}:{\\quad}\\frac{-1-\\sqrt{4x-23}}{2}$$", "input": "\\frac{-1-\\sqrt{4x-23}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-1-\\sqrt{4x-23}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7LcON8lIS+hibufdR4EzXnKLtgQ/buQxiZtxG95g5rqa2yYtEqeT7UjI2zAoMd7PXcJChiVhDxT5N/LHSTLMjyCoNzfyRy0Ar5mHzAcyfgCkh8LHQeR3zCTVr2pbPKm/fJMdNUVWcJhfOcXhVwo2bw1OzjFFhwU2jBC8En0RaGbBBPUDwBnNzGwo5/z2vgwUCbGUhD9+gDQg079F600Xu5Q==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "y=\\frac{-1+\\sqrt{4x-23}}{2},\\:y=\\frac{-1-\\sqrt{4x-23}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "\\frac{-1+\\sqrt{4x-23}}{2},\\:\\frac{-1-\\sqrt{4x-23}}{2}" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x^{2}+x+6" }, "showViewLarger": true } }, "meta": { "showVerify": true } }