domain of f(x)=(sqrt(x-4))/(x-10)

{ "query": { "display": "domain $$f\\left(x\\right)=\\frac{\\sqrt{x-4}}{x-10}$$", "symbolab_question": "FUNCTION#domain f(x)=\\frac{\\sqrt{x-4}}{x-10}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "4\\le x<10\\lor x>10", "interval": "[4,10)\\cup (10,\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\frac{\\sqrt{x-4}}{x-10}\\::{\\quad}4\\le\\:x<10\\lor\\:x>10$$", "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:$${\\quad}x\\ge\\:4$$", "input": "\\frac{\\sqrt{x-4}}{x-10}", "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 $$x-4\\ge\\:0:{\\quad}x\\ge\\:4$$", "input": "x-4\\ge\\:0", "steps": [ { "type": "interim", "title": "Move $$4\\:$$to the right side", "input": "x-4\\ge\\:0", "result": "x\\ge\\:4", "steps": [ { "type": "step", "primary": "Add $$4$$ to both sides", "result": "x-4+4\\ge\\:0+4" }, { "type": "step", "primary": "Simplify", "result": "x\\ge\\:4" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Non Negative Radicals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6uXYXFOxzeSJCgDzUN6d7ekM3cOpJIK7YJJ0SWFBUTA+Zba8vprGflWt+qlqmAkMprsjSyNJPEnDes5YB2uYUOsjAznAh/iXsgzfzTqoJ+d1Kz6wcSjzsEGK62xj66zEuwrfLUdq47TxAxJHOZMKbq0T0hsJCAMR1z7hZos8Z4UL5yXTiQH+xVI8jZizlacbPLCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=10$$", "input": "\\frac{\\sqrt{x-4}}{x-10}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{\\sqrt{x-4}}{x-10}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x-10=0:{\\quad}x=10$$", "input": "x-10=0", "steps": [ { "type": "interim", "title": "Move $$10\\:$$to the right side", "input": "x-10=0", "result": "x=10", "steps": [ { "type": "step", "primary": "Add $$10$$ to both sides", "result": "x-10+10=0+10" }, { "type": "step", "primary": "Simplify", "result": "x=10" } ], "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=10" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6uXYXFOxzeSJCgDzUN6d7ekM3cOpJIK7YJJ0SWFBUTA+Zba8vprGflWt+qlqmAkMphAHn4JE4dHebHgwg5qEoq+ci1/RU7YCZvZqreyS77CUfH6kVvau+ENR7awhAU+mEsp0PSiCnyoEmiGfe1w06kAsTlLa2EauAbzTa9l1LuqO/ixK5hDN38dY2dE4xNHaow==" } }, { "type": "step", "primary": "Combine real regions and undefined points for final function domain", "result": "4\\le\\:x<10\\lor\\:x>10" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{\\sqrt{x-4}}{x-10}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }