domain of f(x)=(1-2t)/((t+6))

{ "query": { "display": "domain $$f\\left(x\\right)=\\frac{1-2t}{\\left(t+6\\right)}$$", "symbolab_question": "FUNCTION#domain f(x)=\\frac{1-2t}{(t+6)}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "t<-6\\lor t>-6", "interval": "(-\\infty ,-6)\\cup (-6,\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\frac{1-2t}{t+6}\\::{\\quad}t<-6\\lor\\:t>-6$$", "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:$${\\quad}t=-6$$", "input": "\\frac{1-2t}{t+6}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{1-2t}{t+6}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$t+6=0:{\\quad}t=-6$$", "input": "t+6=0", "steps": [ { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "t+6=0", "result": "t=-6", "steps": [ { "type": "step", "primary": "Subtract $$6$$ from both sides", "result": "t+6-6=0-6" }, { "type": "step", "primary": "Simplify", "result": "t=-6" } ], "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": "t=-6" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6tXi3rba8gRz9aEIIsojR2dW9IywSqFQpOf5hkY5ANjNmxhhvhGPdXDo331HaiBP/D8v81YuLPtaVdgnKLSZ5mfGGa871gItIFrU3m71GZbJeqXxdc+rps1CUyb7fqI2GeNdMUZEa0oRZcGEhWpGBVTIiZ+gAqC183B3PyLTCJX4iO/UtL46KHKVIVkdaLuvkA==" } }, { "type": "step", "primary": "The function domain", "result": "t<-6\\lor\\:t>-6" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{1-2t}{t+6}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }