domain of sqrt(2x^2-17)

{ "query": { "display": "domain $$\\sqrt{2x^{2}-17}$$", "symbolab_question": "FUNCTION#domain \\sqrt{2x^{2}-17}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "x\\le -\\sqrt{\\frac{17}{2}}\\lor x\\ge \\sqrt{\\frac{17}{2}}", "interval": "(-\\infty ,-\\sqrt{\\frac{17}{2}}]\\cup [\\sqrt{\\frac{17}{2}},\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\sqrt{2x^{2}-17}\\::{\\quad}x\\le\\:-\\sqrt{\\frac{17}{2}}\\lor\\:x\\ge\\:\\sqrt{\\frac{17}{2}}$$", "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:$${\\quad}x\\le\\:-\\sqrt{\\frac{17}{2}}\\lor\\:x\\ge\\:\\sqrt{\\frac{17}{2}}$$", "input": "\\sqrt{2x^{2}-17}", "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 $$2x^{2}-17\\ge\\:0:{\\quad}x\\le\\:-\\sqrt{\\frac{17}{2}}\\lor\\:x\\ge\\:\\sqrt{\\frac{17}{2}}$$", "input": "2x^{2}-17\\ge\\:0", "steps": [ { "type": "interim", "title": "Move $$17\\:$$to the right side", "input": "2x^{2}-17\\ge\\:0", "result": "2x^{2}\\ge\\:17", "steps": [ { "type": "step", "primary": "Add $$17$$ to both sides", "result": "2x^{2}-17+17\\ge\\:0+17" }, { "type": "step", "primary": "Simplify", "result": "2x^{2}\\ge\\:17" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x^{2}\\ge\\:17", "result": "x^{2}\\ge\\:\\frac{17}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x^{2}}{2}\\ge\\:\\frac{17}{2}" }, { "type": "step", "primary": "Simplify", "result": "x^{2}\\ge\\:\\frac{17}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "For $$u^n\\:\\ge\\:\\:a$$, if $$n\\:$$is even then $$u\\:\\le\\:\\:-\\sqrt[n]{a}\\:or\\:u\\:\\ge\\:\\:\\sqrt[n]{a}$$" }, { "type": "step", "result": "x\\le\\:-\\sqrt{\\frac{17}{2}}\\lor\\:x\\ge\\:\\sqrt{\\frac{17}{2}}" } ], "meta": { "solvingClass": "Inequalities", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Non Negative Radicals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6pQrXTqjCbfnTUAjMcCJHWDTnzHD16aDvA9/DvEtY6Cuiy1x4r5yESkIFR8S3Sx6lER97HwlaFhPhACf+Jbd/h/BO9SHIbS0S2TkMcQNFLo71sD7NfhsPe7eDHrmjY0mEz4+hD29TGjrbAdyB62rRfuiohRtRO+ZuWRNCqxsqeyeSHkHUU84Fe5wwHce8hhvdw==" } }, { "type": "step", "primary": "The function domain", "result": "x\\le\\:-\\sqrt{\\frac{17}{2}}\\lor\\:x\\ge\\:\\sqrt{\\frac{17}{2}}" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sqrt{2x^{2}-17}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }