inverse of f(x)=x^2+2x+4

{ "query": { "display": "inverse $$f\\left(x\\right)=x^{2}+2x+4$$", "symbolab_question": "FUNCTION#inverse f(x)=x^{2}+2x+4" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "-1+\\sqrt{x-3},-\\sqrt{x-3}-1", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$x^{2}+2x+4:{\\quad}-1+\\sqrt{x-3},\\:-\\sqrt{x-3}-1$$", "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}+2x+4" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=x^{2}+2x+4", "result": "x=y^{2}+2y+4", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=y^{2}+2y+4" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZURO4sQOZqjjRUFEev88iQuJDNT2wCif15w2ce06so24ADUha1tXi8qEmnvLTUBI9T/L0MoYg+CUn6oyL3EO7YpV5U35Ldoq28O/2Mp19YBuczb172HvovdBpNcutsB6xUlwt5S0mvTtHzYltVBIUs0=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=y^{2}+2y+4$$", "input": "x=y^{2}+2y+4", "steps": [ { "type": "step", "primary": "Switch sides", "result": "y^{2}+2y+4=x" }, { "type": "interim", "title": "Move $$x\\:$$to the left side", "input": "y^{2}+2y+4=x", "result": "y^{2}+2y+4-x=0", "steps": [ { "type": "step", "primary": "Subtract $$x$$ from both sides", "result": "y^{2}+2y+4-x=x-x" }, { "type": "step", "primary": "Simplify", "result": "y^{2}+2y+4-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}+2y+4-x=0", "result": "{y}_{1,\\:2}=\\frac{-2\\pm\\:\\sqrt{2^{2}-4\\cdot\\:1\\cdot\\:\\left(4-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 $${\\quad}a=1,\\:b=2,\\:c=4-x$$", "result": "{y}_{1,\\:2}=\\frac{-2\\pm\\:\\sqrt{2^{2}-4\\cdot\\:1\\cdot\\:\\left(4-x\\right)}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jcj3Xaxo+viXN2ZB0Y3z7rHuOC2hZ+uVk9puSrhnYBpncagboqjyMM+Hi/KaGzT6++OtdadYzsAJ/iG5FNZZmNIngK9cKGwUTL8Xgn6paPXl5PkSbCe1DHfYsip3OZP2bZ5nCZNlu3FWGZ9lKe02vF0TOyStaUIiJ678Y9mwOCeXLZ3nio7BU7VZyCWaD0BLQMMORWV+KB+bs00UHfa3AIktuuzL09SS3U+RsgxHGFXM/LAqhVuCkTIuc6bWnflIrKAIgmS8W9timVbRiFNYjf+bMQ5k1Tw2+UqHAExFjviOdUefcwyUioepLb1LThT7AoNo+4la/A9M5pnz4tMsOmYJTz9iY6R2JNTfPX0KV5u2+9vzrtnl6+0mcvSNieKFbdNl//QVuJjXX1DbjBkNuZiWk9GMxTHCKLlF4R6hOQged9TIA/KnqmGBBYRLrKTm0W3AdxChn1fX7F/ZLSj3eUkot7PcZS/Y6l8t49dHfmrWwPs1+Gw97t4MeuaNjSYTwFThB5UvC5hv25aXOXZInxpZf3MZuUW2giNK0kv8UXn4X+gLrVT1QnUPUlWG+xN1" } }, { "type": "interim", "title": "Simplify $$\\sqrt{2^{2}-4\\cdot\\:1\\cdot\\:\\left(4-x\\right)}:{\\quad}2\\sqrt{x-3}$$", "input": "\\sqrt{2^{2}-4\\cdot\\:1\\cdot\\:\\left(4-x\\right)}", "result": "{y}_{1,\\:2}=\\frac{-2\\pm\\:2\\sqrt{x-3}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=\\sqrt{2^{2}-4\\left(-x+4\\right)}" }, { "type": "interim", "title": "Factor $$2^{2}-4\\left(4-x\\right):{\\quad}4\\left(x-3\\right)$$", "input": "2^{2}-4\\left(4-x\\right)", "result": "=\\sqrt{4\\left(x-3\\right)}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=4\\cdot\\:1-4\\left(4-x\\right)" }, { "type": "step", "primary": "Factor out common term $$4$$", "result": "=4\\left(1-\\left(4-x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Expand $$-\\left(-x+4\\right)+1:{\\quad}x-3$$", "input": "1-\\left(4-x\\right)", "result": "=4\\left(x-3\\right)", "steps": [ { "type": "interim", "title": "$$-\\left(4-x\\right):{\\quad}-4+x$$", "input": "-\\left(4-x\\right)", "result": "=1-4+x", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-4-\\left(-x\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-4+x" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Subtract the numbers: $$1-4=-3$$", "result": "=x-3" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77bZl4MpztTi37zMif9EpnQOfOVs9mPIqDLV5QIWwt3n2p3ejuBGDFkG7AXaB5hm11sD7NfhsPe7eDHrmjY0mEwiamWt0P/cARJHquCMzYeMOpMusR54rMW7EyUFw8Noa" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\sqrt{4}\\sqrt{x-3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=2\\sqrt{x-3}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Separate the solutions", "result": "{y}_{1}=\\frac{-2+2\\sqrt{x-3}}{2\\cdot\\:1},\\:{y}_{2}=\\frac{-2-2\\sqrt{x-3}}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$y=\\frac{-2+2\\sqrt{x-3}}{2\\cdot\\:1}:{\\quad}-1+\\sqrt{x-3}$$", "input": "\\frac{-2+2\\sqrt{x-3}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-2+2\\sqrt{x-3}}{2}" }, { "type": "interim", "title": "Factor $$-2+2\\sqrt{x-3}:{\\quad}2\\left(-1+\\sqrt{-3+x}\\right)$$", "input": "-2+2\\sqrt{x-3}", "result": "=\\frac{2\\left(-1+\\sqrt{-3+x}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=-2\\cdot\\:1+2\\sqrt{-3+x}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(-1+\\sqrt{-3+x}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=-1+\\sqrt{x-3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qdM0Yoz2jn7FcqfuJhCvnSEIt9L/MzYuRjNxa6w5ryh36K2zeHQvt/hJP3xKS9q6zMFYmi1F5Hg/ibpEToVnYwVfNSZqHw/6MPoUbcIPpVjSQE/E9NXgeamgpIXRf+E6FOSncCxzneMWT7EsQpWhue55x4AkZVY3DAhgEMQdVdXX9K7lYKjT8/24FH1bkA/1" } }, { "type": "interim", "title": "$$y=\\frac{-2-2\\sqrt{x-3}}{2\\cdot\\:1}:{\\quad}-\\sqrt{x-3}-1$$", "input": "\\frac{-2-2\\sqrt{x-3}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-2-2\\sqrt{x-3}}{2}" }, { "type": "interim", "title": "Factor $$-2-2\\sqrt{x-3}:{\\quad}-2\\left(1+\\sqrt{-3+x}\\right)$$", "input": "-2-2\\sqrt{x-3}", "result": "=-\\frac{2\\left(1+\\sqrt{-3+x}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=-2\\cdot\\:1-2\\sqrt{-3+x}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=-2\\left(1+\\sqrt{-3+x}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=-\\left(\\sqrt{x-3}+1\\right)" }, { "type": "step", "primary": "Negate $$-\\left(\\sqrt{x-3}+1\\right)=-\\sqrt{x-3}-1$$", "result": "=-\\sqrt{x-3}-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ey5bhjW0qEToBYarHkykkSEIt9L/MzYuRjNxa6w5ryh36K2zeHQvt/hJP3xKS9q6zMFYmi1F5Hg/ibpEToVnYzQBr2DXqehsX6/YwO9YZN3SQE/E9NXgeamgpIXRf+E6QcW9Y2e0g1HCM12AQZHyHe55x4AkZVY3DAhgEMQdVdXX9K7lYKjT8/24FH1bkA/1" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "y=-1+\\sqrt{x-3},\\:y=-\\sqrt{x-3}-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "-1+\\sqrt{x-3},\\:-\\sqrt{x-3}-1" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x^{2}+2x+4" }, "showViewLarger": true } }, "meta": { "showVerify": true } }