inverse of f(x)=(x+3)^2-2

{ "query": { "display": "inverse $$f\\left(x\\right)=\\left(x+3\\right)^{2}-2$$", "symbolab_question": "FUNCTION#inverse f(x)=(x+3)^{2}-2" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\sqrt{x+2}-3,-\\sqrt{x+2}-3", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$\\left(x+3\\right)^{2}-2:{\\quad}\\sqrt{x+2}-3,\\:-\\sqrt{x+2}-3$$", "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=\\left(x+3\\right)^{2}-2" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=\\left(x+3\\right)^{2}-2", "result": "x=\\left(y+3\\right)^{2}-2", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=\\left(y+3\\right)^{2}-2" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZdY7iMyUU3WTUlVAKVk9b4b4KqpqHsTkSBgy/axoI6DamTAUhNuvP+oYRx2ZQG6x18Q3tfXX1jSRYh4EFIfu+U5DNOwm/hV0YV0QkO8wKai7EaTcoaJcD/yzWbRAAledhROjRxpO2XYmGPYDdnsqYeQ=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=\\left(y+3\\right)^{2}-2$$", "input": "x=\\left(y+3\\right)^{2}-2", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\left(y+3\\right)^{2}-2=x" }, { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "\\left(y+3\\right)^{2}-2=x", "result": "\\left(y+3\\right)^{2}=x+2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "\\left(y+3\\right)^{2}-2+2=x+2" }, { "type": "step", "primary": "Simplify", "result": "\\left(y+3\\right)^{2}=x+2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "For $$\\left(g\\left(x\\right)\\right)^{2}=f\\left(a\\right)$$ the solutions are $$g\\left(x\\right)=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "interim", "title": "Solve $$y+3=\\sqrt{x+2}:{\\quad}y=\\sqrt{x+2}-3$$", "input": "y+3=\\sqrt{x+2}", "steps": [ { "type": "interim", "title": "Move $$3\\:$$to the right side", "input": "y+3=\\sqrt{x+2}", "result": "y=\\sqrt{x+2}-3", "steps": [ { "type": "step", "primary": "Subtract $$3$$ from both sides", "result": "y+3-3=\\sqrt{x+2}-3" }, { "type": "step", "primary": "Simplify", "result": "y=\\sqrt{x+2}-3" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$y+3=-\\sqrt{x+2}:{\\quad}y=-\\sqrt{x+2}-3$$", "input": "y+3=-\\sqrt{x+2}", "steps": [ { "type": "interim", "title": "Move $$3\\:$$to the right side", "input": "y+3=-\\sqrt{x+2}", "result": "y=-\\sqrt{x+2}-3", "steps": [ { "type": "step", "primary": "Subtract $$3$$ from both sides", "result": "y+3-3=-\\sqrt{x+2}-3" }, { "type": "step", "primary": "Simplify", "result": "y=-\\sqrt{x+2}-3" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "y=\\sqrt{x+2}-3,\\:y=-\\sqrt{x+2}-3" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "\\sqrt{x+2}-3,\\:-\\sqrt{x+2}-3" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "(x+3)^{2}-2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }