domain of f(x)= 1/(sqrt(6x-12))

{ "query": { "display": "domain $$f\\left(x\\right)=\\frac{1}{\\sqrt{6x-12}}$$", "symbolab_question": "FUNCTION#domain f(x)=\\frac{1}{\\sqrt{6x-12}}" }, "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 $$\\frac{1}{\\sqrt{6x-12}}\\::{\\quad}x>2$$", "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{1}{\\sqrt{6x-12}}", "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 $$6x-12\\ge\\:0:{\\quad}x\\ge\\:2$$", "input": "6x-12\\ge\\:0", "steps": [ { "type": "interim", "title": "Move $$12\\:$$to the right side", "input": "6x-12\\ge\\:0", "result": "6x\\ge\\:12", "steps": [ { "type": "step", "primary": "Add $$12$$ to both sides", "result": "6x-12+12\\ge\\:0+12" }, { "type": "step", "primary": "Simplify", "result": "6x\\ge\\:12" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6x\\ge\\:12", "result": "x\\ge\\:2", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6x}{6}\\ge\\:\\frac{12}{6}" }, { "type": "step", "primary": "Simplify", "result": "x\\ge\\:2" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Non Negative Radicals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6sAk/vs29I8QjCu0bMiO0lwcuU5X8vgvIIlcEXIxsgAKFsqbNlWc0w7a+8NXBmI+EQoG1kCQe4jjK4hlqcH8KoPHaWUDHumUntWgiXRHSU10P8vQyhiD4JSfqjIvcQ7tighNdTNhkC/HNgXzAXj97g6Dx5JKmKhgZnY8qot6mD7yL4KM2VvnpXkeFLvfz83/bb8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=2$$", "input": "\\frac{1}{\\sqrt{6x-12}}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{1}{\\sqrt{6x-12}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$\\sqrt{6x-12}=0:{\\quad}x=2$$", "input": "\\sqrt{6x-12}=0", "steps": [ { "type": "interim", "title": "Square both sides:$${\\quad}6x-12=0$$", "input": "\\sqrt{6x-12}=0", "result": "6x-12=0", "steps": [ { "type": "step", "result": "\\left(\\sqrt{6x-12}\\right)^{2}=0^{2}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt{6x-12}\\right)^{2}:{\\quad}6x-12$$", "input": "\\left(\\sqrt{6x-12}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(6x-12\\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(6x-12\\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": "=6x-12", "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+eNAiQR2GWybmYZoFXPSioPMwJ2+sC8ksde3EL3dW7N7EScc6upJ8tRrcqdzpqw4uWy22Tu/0m71n2FfPReH2RI2dnj8bYA0b6V2RSTOZ7Os9NOD0798xTKV1eCCGmlDC3mMW2tTVmd09XElMYIMJjAj0C4=" } }, { "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": "6x-12=0" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDaZCNCMlwFw9Zrpnfgo8QWVRjG/ww5ZpLfKK0xrHkfkRisOsUeHnX6OraBu6ek9MdM2S9j8g/a7SZ7JmR2xSSaEOqTZD+VhbX2q5Z5JnuKRzbccY9icqzoj+/Fx/GgQ8I0wAcA81xxkOtKBqQJLHq2sIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Solve $$6x-12=0:{\\quad}x=2$$", "input": "6x-12=0", "steps": [ { "type": "interim", "title": "Move $$12\\:$$to the right side", "input": "6x-12=0", "result": "6x=12", "steps": [ { "type": "step", "primary": "Add $$12$$ to both sides", "result": "6x-12+12=0+12" }, { "type": "step", "primary": "Simplify", "result": "6x=12" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6x=12", "result": "x=2", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6x}{6}=\\frac{12}{6}" }, { "type": "step", "primary": "Simplify", "result": "x=2" } ], "meta": { "interimType": "Divide Both Sides Specific 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 $$\\sqrt{6x-12}=0$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug in $$x=2:{\\quad}$$True", "input": "\\sqrt{6\\cdot\\:2-12}=0", "steps": [ { "type": "interim", "title": "$$\\sqrt{6\\cdot\\:2-12}=0$$", "input": "\\sqrt{6\\cdot\\:2-12}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:2=12$$", "result": "=\\sqrt{12-12}" }, { "type": "step", "primary": "Subtract the numbers: $$12-12=0$$", "result": "=\\sqrt{0}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{0}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lLuv/wUtOE3/CcpTUwIOwHMnFEvKpkTozQN/IewOBsOrju+5Z51e/ZZSD3gRHwjBpin7Xy6oBesxOkuA5k+rpDgmdI4C3OVUWPPdComFRjrhxinwG6T6JLxgba+iOJf3" } }, { "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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6sAk/vs29I8QjCu0bMiO0lwcuU5X8vgvIIlcEXIxsgAKFsqbNlWc0w7a+8NXBmI+EQmrRByO43YpSZTDJd7gFoNCeV58zUtPMYeoSN/G1DKgZEt3ZXAiqUE0HIXrrrezJLv0h/Bl6zflydMyidJivhqsomww5CJusLJKJ/8yCYSP76H0+yuciVWG+klb0oDN0Q==" } }, { "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{1}{\\sqrt{6x-12}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }