domain of (2x+1)/(x-1)

{ "query": { "display": "domain $$\\frac{2x+1}{x-1}$$", "symbolab_question": "FUNCTION#domain \\frac{2x+1}{x-1}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "x<1\\lor x>1", "interval": "(-\\infty ,1)\\cup (1,\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\frac{2x+1}{x-1}\\::{\\quad}x<1\\lor\\:x>1$$", "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 undefined (singularity) points:$${\\quad}x=1$$", "input": "\\frac{2x+1}{x-1}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{2x+1}{x-1}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x-1=0:{\\quad}x=1$$", "input": "x-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1=0", "result": "x=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "x=1" } ], "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 following points are undefined", "result": "x=1" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6qeleDCPJEbWT6Sefn16GIgmKC85IyXQZuYnMoY1uq+DmxhhvhGPdXDo331HaiBP/D8v81YuLPtaVdgnKLSZ5mfGGa871gItIFrU3m71GZbJeqXxdc+rps1CUyb7fqI2GeNdMUZEa0oRZcGEhWpGBVTIiZ+gAqC183B3PyLTCJX4iO/UtL46KHKVIVkdaLuvkA==" } }, { "type": "step", "primary": "The function domain", "result": "x<1\\lor\\:x>1" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{2x+1}{x-1}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }