domain of 9/(sqrt(x+2))

{ "query": { "display": "domain $$\\frac{9}{\\sqrt{x+2}}$$", "symbolab_question": "FUNCTION#domain \\frac{9}{\\sqrt{x+2}}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "x>-2", "interval": "(-2,\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\frac{9}{\\sqrt{x+2}}\\::{\\quad}x>-2$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "interim", "title": "Find non-negative values for radicals:$${\\quad}x\\ge\\:-2$$", "input": "\\frac{9}{\\sqrt{x+2}}", "steps": [ { "type": "step", "primary": "$$\\sqrt{f\\left(x\\right)}\\quad\\Rightarrow\\quad\\:f\\left(x\\right)\\ge{0}\\:$$", "meta": { "general_rule": { "extension": "$$\\sqrt{f\\left(x\\right)}$$ (or any even root) has real values only when $$f\\left(x\\right)\\ge\\:0\\:$$" } } }, { "type": "interim", "title": "Solve $$x+2\\ge\\:0:{\\quad}x\\ge\\:-2$$", "input": "x+2\\ge\\:0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x+2\\ge\\:0", "result": "x\\ge\\:-2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "x+2-2\\ge\\:0-2" }, { "type": "step", "primary": "Simplify", "result": "x\\ge\\:-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Non Negative Radicals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6mFF313vhV+ZfjRVoxY1mM/kh5cKiUAWYkU7+s+s/By635xzgnX+qxL4BF0NMpujwoPHkkqYqGBmdjyqi3qYPvJIlFXt4vc8IacsgLvS9FhRo3oe/oyhMy2+1TQhDBd2f2jmv5FsVs6sV6sMNTZCIaGIVH0SfDX9/3i0hzAT28Fr8W2cSy9bn618nmD3cNSDKw==" } }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=-2$$", "input": "\\frac{9}{\\sqrt{x+2}}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{9}{\\sqrt{x+2}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$\\sqrt{x+2}=0:{\\quad}x=-2$$", "input": "\\sqrt{x+2}=0", "steps": [ { "type": "interim", "title": "Square both sides:$${\\quad}x+2=0$$", "input": "\\sqrt{x+2}=0", "result": "x+2=0", "steps": [ { "type": "step", "result": "\\left(\\sqrt{x+2}\\right)^{2}=0^{2}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt{x+2}\\right)^{2}:{\\quad}x+2$$", "input": "\\left(\\sqrt{x+2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(x+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(x+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": "=x+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+eNAiXGr5ZotNPtt0EAu+F01INqJ9HQfN3qOfAB4kBd9UOFDpA7clkQOa7zoZEIDggSG4WRLd2VwIqlBNByF6663syTD7/TDV+MdcR0GdN6+1FDtalh+geyGIHEPBnO3XqS5kompXFf3SOUx+H18qfp3MLg=" } }, { "type": "interim", "title": "Expand $$0^{2}:{\\quad}0$$", "input": "0^{2}", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y+JsXs5Wey1G4iayV5SJuA7afGBEifydK7wCwslogi9F5bLdlaCFshD3v4rLy7EcFU0qZ4XMwhSlm3knFkjDUN7UD16RqFQha281as3i/3hi3DJlF4JtuHZL1PI6pH9O" } }, { "type": "step", "result": "x+2=0" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDaZCNCMlwFw9Zrpnfgo8QWwb0TMHL0UKK8mZb3owaEpqara+52Td5xcUnwkQYXwhQAiVyPqAZh7yDlcQYf8OCnA8cNB8Ab3wAU7PsKnISMSpdru72i1HipIeOo3/2JsX1TmSqfo3P/KqBfD7c9O7QlJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "Solve $$x+2=0:{\\quad}x=-2$$", "input": "x+2=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x+2=0", "result": "x=-2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "x+2-2=0-2" }, { "type": "step", "primary": "Simplify", "result": "x=-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=-2" }, { "type": "interim", "title": "Verify Solutions:$${\\quad}x=-2\\:$$True", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$\\sqrt{x+2}=0$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug in $$x=-2:{\\quad}$$True", "input": "\\sqrt{\\left(-2\\right)+2}=0", "steps": [ { "type": "interim", "title": "$$\\sqrt{\\left(-2\\right)+2}=0$$", "input": "\\sqrt{\\left(-2\\right)+2}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\sqrt{-2+2}" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-2+2=0$$", "result": "=\\sqrt{0}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{0}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gmkWErJ4PxbwyYKX6HyNfQCWKUbvV6WK3fDUgFtg3Q8GS7g7rdYb4CeWQePA28lVo3oe/oyhMy2+1TQhDBd2f0lYpd3OTPsRZM0CTp+IDQYkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "primary": "$$0=0$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "x=-2" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=-2" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6mFF313vhV+ZfjRVoxY1mM/kh5cKiUAWYkU7+s+s/By6M6/W7rkloBMm6OqTQ9UW9CxOUtrYRq4BvNNr2XUu6o5NuEM6hxBraOt7lssmGP08RSpN33oxZMojoqvYhvSJAM7smHY5JYb8vLrutydV3LMVJfwlOggS7xEa0KlmyHCxxq714IG8qF7AXh5s7M1Yyg==" } }, { "type": "step", "primary": "Combine real regions and undefined points for final function domain", "result": "x>-2" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{9}{\\sqrt{x+2}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }