inverse of f(x)=sqrt(3+7x)

{ "query": { "display": "inverse $$f\\left(x\\right)=\\sqrt{3+7x}$$", "symbolab_question": "FUNCTION#inverse f(x)=\\sqrt{3+7x}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\frac{x^{2}-3}{7}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$\\sqrt{3+7x}:{\\quad}\\frac{x^{2}-3}{7}$$", "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+7x}" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=\\sqrt{3+7x}", "result": "x=\\sqrt{3+7y}", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=\\sqrt{3+7y}" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZeQumNCAZ/5ZQRoJCS6RJdmD0yqNoTWLwGuqxO0C0cFVz5Lb2DzoR+2lK1ma7A4yFw7HH89RUTaeN3FUkEAdGm+LD3IXcw26W+fH/nWE+8Jg6uvWjigmDRR62v/uo5/c/6/iRbMPfLQeIqHWmDQDbe8=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=\\sqrt{3+7y}$$", "input": "x=\\sqrt{3+7y}", "steps": [ { "type": "interim", "title": "Square both sides:$${\\quad}x^{2}=3+7y$$", "input": "x=\\sqrt{3+7y}", "result": "x^{2}=3+7y", "steps": [ { "type": "step", "result": "x^{2}=\\left(\\sqrt{3+7y}\\right)^{2}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt{3+7y}\\right)^{2}:{\\quad}3+7y$$", "input": "\\left(\\sqrt{3+7y}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(3+7y\\right)^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=\\left(3+7y\\right)^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3+7y", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vz0OwVmz7FbkxdUH+eNAiVaHHFeGsVlSba/pxL/x6RHptRYmj6WUpgaKrNIresW2wNmnTNPfSXaFwWcdWyIh039H/SFX0n2Ge344m0fgESR0U1L0UOfZXYZXOhi7eOgpoZhmqVEowNjy42o5yO862votFMd5aa71+dAhX5ea6wE=" } }, { "type": "step", "result": "x^{2}=3+7y" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjCBKdOJvtfezbFI4zIaDphv1Sios1Te+I0yK/nO5j0RsmVBAimTWz5/bcyAzDjmFl0bkDgOhbeCghIkNrATUfx+A2dhZsR0jUVxLBB9f34/aYJU5LhhBmS48qGZHNItQ1R4n8NyjCNMOZXuKiEC3nXUvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "Solve $$x^{2}=3+7y:{\\quad}y=\\frac{x^{2}-3}{7}$$", "input": "x^{2}=3+7y", "steps": [ { "type": "step", "primary": "Switch sides", "result": "3+7y=x^{2}" }, { "type": "interim", "title": "Move $$3\\:$$to the right side", "input": "3+7y=x^{2}", "result": "7y=x^{2}-3", "steps": [ { "type": "step", "primary": "Subtract $$3$$ from both sides", "result": "3+7y-3=x^{2}-3" }, { "type": "step", "primary": "Simplify", "result": "7y=x^{2}-3" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$7$$", "input": "7y=x^{2}-3", "result": "y=\\frac{x^{2}-3}{7}", "steps": [ { "type": "step", "primary": "Divide both sides by $$7$$", "result": "\\frac{7y}{7}=\\frac{x^{2}}{7}-\\frac{3}{7}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{7y}{7}=\\frac{x^{2}}{7}-\\frac{3}{7}", "result": "y=\\frac{x^{2}-3}{7}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{7y}{7}:{\\quad}y$$", "input": "\\frac{7y}{7}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{7}{7}=1$$", "result": "=y" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bDSUqLCDxNBkTY388RFpjC061ljBSPJeENOw2efoSWu6Rbu0VBfiSYvGZCQ6uC9JRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6qLCvS8Zea+DTiB98JntyUM" } }, { "type": "interim", "title": "Simplify $$\\frac{x^{2}}{7}-\\frac{3}{7}:{\\quad}\\frac{x^{2}-3}{7}$$", "input": "\\frac{x^{2}}{7}-\\frac{3}{7}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{x^{2}-3}{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HBMy1k3euUkMrxh46KjL3uKaMifbDtyxSiaTnv+pm1ktOtZYwUjyXhDTsNnn6ElrSGpx9e3Lolh3WlYiBDdmOIDiqV9LiwjmI5GzQZHNv+ujeh7+jKEzLb7VNCEMF3Z/bMzoTd+5nEXVeQoBhpFcIKh1rlLVf/UMD1VEgMr4JWx9cxSVspwjErKjMCT/CXZNJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "y=\\frac{x^{2}-3}{7}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "y=\\frac{x^{2}-3}{7}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "\\frac{x^{2}-3}{7}" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sqrt{3+7x}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }