domain of f(x)=(x^2-6x+5)/(x-5)

{ "query": { "display": "domain $$f\\left(x\\right)=\\frac{x^{2}-6x+5}{x-5}$$", "symbolab_question": "FUNCTION#domain f(x)=\\frac{x^{2}-6x+5}{x-5}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "x<5\\lor x>5", "interval": "(-\\infty ,5)\\cup (5,\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\frac{x^{2}-6x+5}{x-5}\\::{\\quad}x<5\\lor\\:x>5$$", "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:$${\\quad}x=5$$", "input": "\\frac{x^{2}-6x+5}{x-5}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{x^{2}-6x+5}{x-5}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x-5=0:{\\quad}x=5$$", "input": "x-5=0", "steps": [ { "type": "interim", "title": "Move $$5\\:$$to the right side", "input": "x-5=0", "result": "x=5", "steps": [ { "type": "step", "primary": "Add $$5$$ to both sides", "result": "x-5+5=0+5" }, { "type": "step", "primary": "Simplify", "result": "x=5" } ], "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=5" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6o5O3pzDRfWNgCdqshzA9RONUnO7FLlJvlsgXRBwijtfqUE1xsatCkHtfQYqoIkv8GFc2nNahNWcQmyfomJnFSUgcFTh+Pg7MRZzT4GP+eQeo3oe/oyhMy2+1TQhDBd2f1/RDt6XPLyjiMit25raJ2P31dqKnPu+VUTS8EKz0JPa1a/fEl9nEq1bCVIo1BGARg==" } }, { "type": "step", "primary": "The function domain", "result": "x<5\\lor\\:x>5" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{x^{2}-6x+5}{x-5}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }