inverse of f(x)=2x^2-4x-5

{ "query": { "display": "inverse $$f\\left(x\\right)=2x^{2}-4x-5$$", "symbolab_question": "FUNCTION#inverse f(x)=2x^{2}-4x-5" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\frac{4+\\sqrt{8x+56}}{4},\\frac{4-\\sqrt{8x+56}}{4}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$2x^{2}-4x-5:{\\quad}\\frac{4+\\sqrt{8x+56}}{4},\\:\\frac{4-\\sqrt{8x+56}}{4}$$", "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=2x^{2}-4x-5" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=2x^{2}-4x-5", "result": "x=2y^{2}-4y-5", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=2y^{2}-4y-5" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZUXa3lpbGeNrDryXYjJUwgL4KqpqHsTkSBgy/axoI6DamTAUhNuvP+oYRx2ZQG6x18Q3tfXX1jSRYh4EFIfu+U5DNOwm/hV0YV0QkO8wKai7EaTcoaJcD/yzWbRAAledhROjRxpO2XYmGPYDdnsqYeQ=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=2y^{2}-4y-5$$", "input": "x=2y^{2}-4y-5", "steps": [ { "type": "step", "primary": "Switch sides", "result": "2y^{2}-4y-5=x" }, { "type": "interim", "title": "Move $$x\\:$$to the left side", "input": "2y^{2}-4y-5=x", "result": "2y^{2}-4y-5-x=0", "steps": [ { "type": "step", "primary": "Subtract $$x$$ from both sides", "result": "2y^{2}-4y-5-x=x-x" }, { "type": "step", "primary": "Simplify", "result": "2y^{2}-4y-5-x=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7naIlrcNPBUMns2vVwwdwtPvuKu7TDcspo46FJVzS7NFst7spUy7En4Aj0Ijvqo0vtQkjlpCsx+54yA1TyhKxGJKDbr9DTe4g0WQvfJTADCmprMcVnKCjwfOrXgt8hZ9PvFZh+W/iDMDAo4iCI7tFUdwpqaFhGel8eqIiHymEdPNIrutd8eSMgUo+HpRyFwAMhPoQuOU0G7RzIZqgABIE3jT9Z7OSXCqq5qrjeeGQyr3SkOmTLX28mJuZQtYxGVtYlZsbz9LiEDF1Dbz/nht2aXkczN78p6sSSiIN47RfWQmPnUcJGp2WiVMMmEnAtUws75VLPbLDeBrnUBvlTj9xp8rfCxUJuVKbG7GeYn05IRRkaSCajGg2VicA6RuUlsHLHe2RukSECEepZtRJJe9Uu0UqTd96MWTKI6Kr2Ib0iQBTC5HQ4vpNwmK4y2JaM+mDQPxx0GDfWu/Vong9ymSaD58xiBz8lrFG/GdFMb9MzcI=" } }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "2y^{2}-4y-5-x=0", "result": "{y}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:\\sqrt{\\left(-4\\right)^{2}-4\\cdot\\:2\\left(-5-x\\right)}}{2\\cdot\\:2}", "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 $${\\quad}a=2,\\:b=-4,\\:c=-5-x$$", "result": "{y}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:\\sqrt{\\left(-4\\right)^{2}-4\\cdot\\:2\\left(-5-x\\right)}}{2\\cdot\\:2}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "Simplify $$\\sqrt{\\left(-4\\right)^{2}-4\\cdot\\:2\\left(-5-x\\right)}:{\\quad}\\sqrt{8x+56}$$", "input": "\\sqrt{\\left(-4\\right)^{2}-4\\cdot\\:2\\left(-5-x\\right)}", "result": "{y}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:\\sqrt{8x+56}}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-4\\right)^{2}=4^{2}$$" ], "result": "=\\sqrt{4^{2}-4\\cdot\\:2\\left(-x-5\\right)}" }, { "type": "step", "primary": "Refine", "result": "=\\sqrt{16-8\\left(-x-5\\right)}" }, { "type": "interim", "title": "Expand $$16-8\\left(-5-x\\right):{\\quad}8x+56$$", "input": "16-8\\left(-5-x\\right)", "result": "=\\sqrt{8x+56}", "steps": [ { "type": "interim", "title": "Expand $$-8\\left(-5-x\\right):{\\quad}40+8x$$", "input": "-8\\left(-5-x\\right)", "result": "=16+40+8x", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=-8,\\:b=-5,\\:c=x$$" ], "result": "=-8\\left(-5\\right)-\\left(-8\\right)x", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a$$" ], "result": "=8\\cdot\\:5+8x" }, { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:5=40$$", "result": "=40+8x" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EY5dAWHDmMJsHs8FU+3YmM0ag8T1MwTer44+aCS/ZFBKNMG228bk1YvoxYZt2aCy72wZm7kDUxdE6YSmfEbr2nDNBc+dJaAlTRWElvb6AprPvNhHhMcCWdi7z9pbOWVz" } }, { "type": "step", "primary": "Add the numbers: $$16+40=56$$", "result": "=8x+56" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MW7r6fzleSFUnoTdAH8viiAn9lkDfZkicUGkO3EF+IqCQt27SS98bsiPxy9tdA+1o3oe/oyhMy2+1TQhDBd2f3FGtZiGdBdn/vMAKrUamk+S83ontVFh5yYto+EgJwSs" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Separate the solutions", "result": "{y}_{1}=\\frac{-\\left(-4\\right)+\\sqrt{8x+56}}{2\\cdot\\:2},\\:{y}_{2}=\\frac{-\\left(-4\\right)-\\sqrt{8x+56}}{2\\cdot\\:2}" }, { "type": "interim", "title": "$$y=\\frac{-\\left(-4\\right)+\\sqrt{8x+56}}{2\\cdot\\:2}:{\\quad}\\frac{4+\\sqrt{8x+56}}{4}$$", "input": "\\frac{-\\left(-4\\right)+\\sqrt{8x+56}}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{4+\\sqrt{8x+56}}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{4+\\sqrt{8x+56}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EgNtd2sSSbiVF1lgYPDeNWQyKVtKYwTRH7WTSvTe/Oa1yJwPLIjq8q9mwBFWPwFtdYPfXQvX4/bINBB8wSEQ0XT1uYtOjR1Cx40zKro8gGaoN1lH/b7kDiX5kgdkSoaC0kBPxPTV4HmpoKSF0X/hOll7QwFfQ0/2doI0lTAUXIdsLIys5qnuJXg0yObYsrC5fHoXIQE40FG2d5xHqpEqgg==" } }, { "type": "interim", "title": "$$y=\\frac{-\\left(-4\\right)-\\sqrt{8x+56}}{2\\cdot\\:2}:{\\quad}\\frac{4-\\sqrt{8x+56}}{4}$$", "input": "\\frac{-\\left(-4\\right)-\\sqrt{8x+56}}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{4-\\sqrt{8x+56}}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{4-\\sqrt{8x+56}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X80L5EAKoSnk6VOej4yYeWQyKVtKYwTRH7WTSvTe/Oa1yJwPLIjq8q9mwBFWPwFtdYPfXQvX4/bINBB8wSEQ0ed6qN2posnASdE34MVt9BGoN1lH/b7kDiX5kgdkSoaC0kBPxPTV4HmpoKSF0X/hOgth2pMqOPNQLUDaMPPpHJZsLIys5qnuJXg0yObYsrC5fHoXIQE40FG2d5xHqpEqgg==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "y=\\frac{4+\\sqrt{8x+56}}{4},\\:y=\\frac{4-\\sqrt{8x+56}}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "\\frac{4+\\sqrt{8x+56}}{4},\\:\\frac{4-\\sqrt{8x+56}}{4}" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2x^{2}-4x-5" }, "showViewLarger": true } }, "meta": { "showVerify": true } }