inverse of f(x)=7x^2-10

{ "query": { "display": "inverse $$f\\left(x\\right)=7x^{2}-10$$", "symbolab_question": "FUNCTION#inverse f(x)=7x^{2}-10" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\sqrt{\\frac{x+10}{7}},-\\sqrt{\\frac{x+10}{7}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$7x^{2}-10:{\\quad}\\sqrt{\\frac{x+10}{7}},\\:-\\sqrt{\\frac{x+10}{7}}$$", "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=7x^{2}-10" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=7x^{2}-10", "result": "x=7y^{2}-10", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=7y^{2}-10" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZZC3l6r6JMUWD3yEnVb3AmMRbanK99cKyYwa/mBTZldqH9GE5izqXI/bDuw3oLXKo2RLd2VwIqlBNByF6663syTDkDkXF8vCx6Q9CRnz+NFaVBWbEKIE9nJ8YkcdZveNBKw7n5+P/GePwavH269Q9eA=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=7y^{2}-10$$", "input": "x=7y^{2}-10", "steps": [ { "type": "step", "primary": "Switch sides", "result": "7y^{2}-10=x" }, { "type": "interim", "title": "Move $$10\\:$$to the right side", "input": "7y^{2}-10=x", "result": "7y^{2}=x+10", "steps": [ { "type": "step", "primary": "Add $$10$$ to both sides", "result": "7y^{2}-10+10=x+10" }, { "type": "step", "primary": "Simplify", "result": "7y^{2}=x+10" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$7$$", "input": "7y^{2}=x+10", "result": "y^{2}=\\frac{x+10}{7}", "steps": [ { "type": "step", "primary": "Divide both sides by $$7$$", "result": "\\frac{7y^{2}}{7}=\\frac{x}{7}+\\frac{10}{7}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{7y^{2}}{7}=\\frac{x}{7}+\\frac{10}{7}", "result": "y^{2}=\\frac{x+10}{7}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{7y^{2}}{7}:{\\quad}y^{2}$$", "input": "\\frac{7y^{2}}{7}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{7}{7}=1$$", "result": "=y^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AzgCUa1kY/+qseFTJN+Kz1LKqZZl61cSoVEUbXVe4hCjkVi15I8rBefLi4Iyt2wrNBWhdAGWOll7QYcteRzyj4saY2cs9YZ/08feWmaOgkhCm8FAaPiTO4bQo7tK05Pw1nVyonzaAjaqZzCXXQhZOQ==" } }, { "type": "interim", "title": "Simplify $$\\frac{x}{7}+\\frac{10}{7}:{\\quad}\\frac{x+10}{7}$$", "input": "\\frac{x}{7}+\\frac{10}{7}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{x+10}{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NimT6qtleAvRjY1p8aCRDDxL5kXVtNMnIJSm1bo3PNcgJ/ZZA32ZInFBpDtxBfiKXYGCmiBF99lesmXZ9iIfJwMyPQZUpZXj7gdvHuJzylx6pfF1z6umzUJTJvt+ojYZY4LVEbDWfJhdqtpUA6/Oh+3axd5us3DYwfEmTR86iXssurNd+VGAiajE0XcWooEf" } }, { "type": "step", "result": "y^{2}=\\frac{x+10}{7}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7P/muQdRqfqj6lwoMUt8honebbH45oG85ird84sPUJL1Pnie/F7TDTqTxkucfXTe/MLNK6uKFQAPyOS4/KSXEDEy90ptxu5EP0jVXZuHaJw4lpyAwKChPCFtgLUU7AYpvrTH6CH2AfoyKEr0+/Uxe+cZoxkl7XKCNfz0BLTb8L1fdFpY8+TJQURh2lEt0g234z8yxYqOMzSxof2hX/JjCVJRani7UcgHdb0dM6ne4j4BkDsL8Ef9Y86SL0FbKo80Y0bPhaZbYXJs/20C2S2l7Jk8YrmSRejouV6cFhNwTUMDU3KAY4kDHilM6r45aMYmhRT6FvIpi+SIc8+o3gR8t1Ytduin5SBlz49E+ukld6ZgCJ1lQ1FAwVjR6mDWtuTH5DgDlk1i/4yAtEn/PcFV4wKN6Hv6MoTMtvtU0IQwXdn8Ime+VHyE1xbRKHrZpOGTAsIGgq+oYNrCmW/54//1TuiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "y=\\sqrt{\\frac{x+10}{7}},\\:y=-\\sqrt{\\frac{x+10}{7}}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "\\sqrt{\\frac{x+10}{7}},\\:-\\sqrt{\\frac{x+10}{7}}" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "7x^{2}-10" }, "showViewLarger": true } }, "meta": { "showVerify": true } }