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

{ "query": { "display": "inverse $$f\\left(x\\right)=\\left(x^{2}\\right)-4$$", "symbolab_question": "FUNCTION#inverse f(x)=(x^{2})-4" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\sqrt{x+4},-\\sqrt{x+4}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$x^{2}-4:{\\quad}\\sqrt{x+4},\\:-\\sqrt{x+4}$$", "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=x^{2}-4" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=x^{2}-4", "result": "x=y^{2}-4", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=y^{2}-4" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZQSeC8/DQoJkvbgQCKUSzd6ODBqG0bRA4uqHPsV5PG7RRphnRUpbfJBjcr0sJwFNS0UqTd96MWTKI6Kr2Ib0iQCBEx4g4qN8zWEvXnzxyqLfLA8y6avxYBMhBIyqA2CEODiXOJb7o9HjRRcvm85H0Tc=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=y^{2}-4$$", "input": "x=y^{2}-4", "steps": [ { "type": "step", "primary": "Switch sides", "result": "y^{2}-4=x" }, { "type": "interim", "title": "Move $$4\\:$$to the right side", "input": "y^{2}-4=x", "result": "y^{2}=x+4", "steps": [ { "type": "step", "primary": "Add $$4$$ to both sides", "result": "y^{2}-4+4=x+4" }, { "type": "step", "primary": "Simplify", "result": "y^{2}=x+4" } ], "meta": { "interimType": "Move to the Right Title 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+4},\\:y=-\\sqrt{x+4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "\\sqrt{x+4},\\:-\\sqrt{x+4}" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x^{2}-4" }, "showViewLarger": true } }, "meta": { "showVerify": true } }