inverse of f(x)=sqrt(100-x^2)

{ "query": { "display": "inverse $$f\\left(x\\right)=\\sqrt{100-x^{2}}$$", "symbolab_question": "FUNCTION#inverse f(x)=\\sqrt{100-x^{2}}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\sqrt{-x^{2}+100},-\\sqrt{-x^{2}+100}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$\\sqrt{100-x^{2}}:{\\quad}\\sqrt{-x^{2}+100},\\:-\\sqrt{-x^{2}+100}$$", "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{100-x^{2}}" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=\\sqrt{100-x^{2}}", "result": "x=\\sqrt{100-y^{2}}", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=\\sqrt{100-y^{2}}" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZaoVwd8TYgYq19zXFURccSFdjVTVNfTJtGPIDKm6IXF/GQmYVIcZpIrwEyqT9Afp8/wjxGUyXxWBxRqXMq0bSOpN5Aod6Hr1Lp2e/29KhSgUmjqleygtlSszoezKDjz2k7GJwpZAgCxm4WPlcU4xj5k=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=\\sqrt{100-y^{2}}$$", "input": "x=\\sqrt{100-y^{2}}", "steps": [ { "type": "interim", "title": "Square both sides:$${\\quad}x^{2}=100-y^{2}$$", "input": "x=\\sqrt{100-y^{2}}", "result": "x^{2}=100-y^{2}", "steps": [ { "type": "step", "result": "x^{2}=\\left(\\sqrt{100-y^{2}}\\right)^{2}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt{100-y^{2}}\\right)^{2}:{\\quad}100-y^{2}$$", "input": "\\left(\\sqrt{100-y^{2}}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(100-y^{2}\\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(100-y^{2}\\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": "=100-y^{2}", "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+eNAiTy6J1Fx9EROiRk+z4GI+2x9WY+x1CE1XPI0qZBLPyqKs7PdN8ocStDSJ08Eeyb0kcjP9vZe0h5cDZXk0KEZ9KgScn/M2sLYBGgxClqzCxM2ElbEIsL4pGwUEUCMHxaOrZKbYRHzfVesVdOa3X0tw4E=" } }, { "type": "step", "result": "x^{2}=100-y^{2}" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjCBKdOJvtfezbFI4zIaDphvaSELX66rMGeIluOrGG1AErj1sE1rUYarwF0JCJubFAnlrcgJqOukQzeJi1Dvu0yZW8mu0fI0c81GiU/PCqBoptbA+zX4bD3u3gx65o2NJhMqlf5BqLOUUae5S6OHIDTm9cHWhzPInWR2UNxcb0GmMA==" } }, { "type": "interim", "title": "Solve $$x^{2}=100-y^{2}:{\\quad}y=\\sqrt{-x^{2}+100},\\:y=-\\sqrt{-x^{2}+100}$$", "input": "x^{2}=100-y^{2}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "100-y^{2}=x^{2}" }, { "type": "interim", "title": "Move $$100\\:$$to the right side", "input": "100-y^{2}=x^{2}", "result": "-y^{2}=x^{2}-100", "steps": [ { "type": "step", "primary": "Subtract $$100$$ from both sides", "result": "100-y^{2}-100=x^{2}-100" }, { "type": "step", "primary": "Simplify", "result": "-y^{2}=x^{2}-100" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-y^{2}=x^{2}-100", "result": "y^{2}=-x^{2}+100", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-y^{2}}{-1}=\\frac{x^{2}}{-1}-\\frac{100}{-1}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-y^{2}}{-1}=\\frac{x^{2}}{-1}-\\frac{100}{-1}", "result": "y^{2}=-x^{2}+100", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-y^{2}}{-1}:{\\quad}y^{2}$$", "input": "\\frac{-y^{2}}{-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{y^{2}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=y^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YFGp1az+lRaDlVDZCgvzfzFSpzwRDPIFcbKdvhSPhgvMwViaLUXkeD+JukROhWdjYRkUGLdhVqpOTOvmcth9/R429vuTSxWa7B/X3D1oP03AWQmX+FAZQ57eQ8HwbCJCsZMDOItgsQ3X2g7oQAhBrg==" } }, { "type": "interim", "title": "Simplify $$\\frac{x^{2}}{-1}-\\frac{100}{-1}:{\\quad}-x^{2}+100$$", "input": "\\frac{x^{2}}{-1}-\\frac{100}{-1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{x^{2}-100}{-1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{x^{2}-100}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-\\left(x^{2}-100\\right)" }, { "type": "step", "primary": "Distribute parentheses", "result": "=-x^{2}-\\left(-100\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-x^{2}+100" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HBMy1k3euUkMrxh46KjL3tMQKYAEWd2+Wt1YegarsQaoiUyUJEUZ4GZdO626o3guo5FYteSPKwXny4uCMrdsK6WYhly7/aPIaEWGYm74uM3wt9LEn7QCBUukJKctfSJKm5KWrv5y7TlmVxX/22cMl7XbseR8OKG2AyJ/HilxvtUizo7J5mucxlgpfTrnahv6JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "y^{2}=-x^{2}+100" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "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{-x^{2}+100},\\:y=-\\sqrt{-x^{2}+100}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "y=\\sqrt{-x^{2}+100},\\:y=-\\sqrt{-x^{2}+100}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "\\sqrt{-x^{2}+100},\\:-\\sqrt{-x^{2}+100}" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sqrt{100-x^{2}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }