domain of 1/(x^2-10x+15)

{ "query": { "display": "domain $$\\frac{1}{x^{2}-10x+15}$$", "symbolab_question": "FUNCTION#domain \\frac{1}{x^{2}-10x+15}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "x<5-\\sqrt{10}\\lor 5-\\sqrt{10}<x<5+\\sqrt{10}\\lor x>5+\\sqrt{10}", "interval": "(-\\infty ,5-\\sqrt{10})\\cup (5-\\sqrt{10},5+\\sqrt{10})\\cup (5+\\sqrt{10},\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\frac{1}{x^{2}-10x+15}\\::{\\quad}x<5-\\sqrt{10}\\lor\\:5-\\sqrt{10}<x<5+\\sqrt{10}\\lor\\:x>5+\\sqrt{10}$$", "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 undefined (singularity) points:$${\\quad}x=5+\\sqrt{10},\\:x=5-\\sqrt{10}$$", "input": "\\frac{1}{x^{2}-10x+15}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{1}{x^{2}-10x+15}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{2}-10x+15=0:{\\quad}x=5+\\sqrt{10},\\:x=5-\\sqrt{10}$$", "input": "x^{2}-10x+15=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "x^{2}-10x+15=0", "result": "{x}_{1,\\:2}=\\frac{-\\left(-10\\right)\\pm\\:\\sqrt{\\left(-10\\right)^{2}-4\\cdot\\:1\\cdot\\:15}}{2\\cdot\\:1}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=1,\\:b=-10,\\:c=15$$", "result": "{x}_{1,\\:2}=\\frac{-\\left(-10\\right)\\pm\\:\\sqrt{\\left(-10\\right)^{2}-4\\cdot\\:1\\cdot\\:15}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-10\\right)^{2}-4\\cdot\\:1\\cdot\\:15}=2\\sqrt{10}$$", "input": "\\sqrt{\\left(-10\\right)^{2}-4\\cdot\\:1\\cdot\\:15}", "result": "{x}_{1,\\:2}=\\frac{-\\left(-10\\right)\\pm\\:2\\sqrt{10}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-10\\right)^{2}=10^{2}$$" ], "result": "=\\sqrt{10^{2}-4\\cdot\\:1\\cdot\\:15}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:15=60$$", "result": "=\\sqrt{10^{2}-60}" }, { "type": "step", "primary": "$$10^{2}=100$$", "result": "=\\sqrt{100-60}" }, { "type": "step", "primary": "Subtract the numbers: $$100-60=40$$", "result": "=\\sqrt{40}" }, { "type": "interim", "title": "Prime factorization of $$40:{\\quad}2^{3}\\cdot\\:5$$", "input": "40", "result": "=\\sqrt{2^{3}\\cdot\\:5}", "steps": [ { "type": "step", "primary": "$$40\\:$$divides by $$2\\quad\\:40=20\\cdot\\:2$$", "result": "=2\\cdot\\:20" }, { "type": "step", "primary": "$$20\\:$$divides by $$2\\quad\\:20=10\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:10" }, { "type": "step", "primary": "$$10\\:$$divides by $$2\\quad\\:10=5\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:5" }, { "type": "step", "primary": "$$2,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:5" }, { "type": "step", "result": "=2^{3}\\cdot\\:5" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsG/uC0ndYtZpJL4uAxK7FLwPjJo8Fi+uo29zEltAtWEB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIEDm/4ikss/MLnFTWX4cuNoz" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^b\\cdot\\:a^c$$", "result": "=\\sqrt{2^{2}\\cdot\\:2\\cdot\\:5}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{2^{2}}\\sqrt{2\\cdot\\:5}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2\\sqrt{2\\cdot\\:5}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Refine", "result": "=2\\sqrt{10}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xYKyhiwThaKsz2Tl55KkNEFj7CiO8FQNDSt2Nz1mnF7YCxG04c5OKs3H98sf3K9Zq47vuWedXv2WUg94ER8IwfirhMN65g/lcrrrD1WowQqjeh7+jKEzLb7VNCEMF3Z/bMa9W/vMVXpg532eWx0M51MEBIYbd27cthLqV4z2F81K5PcPlwxS+MZ5RLQ+EhleJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{x}_{1}=\\frac{-\\left(-10\\right)+2\\sqrt{10}}{2\\cdot\\:1},\\:{x}_{2}=\\frac{-\\left(-10\\right)-2\\sqrt{10}}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$x=\\frac{-\\left(-10\\right)+2\\sqrt{10}}{2\\cdot\\:1}:{\\quad}5+\\sqrt{10}$$", "input": "\\frac{-\\left(-10\\right)+2\\sqrt{10}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{10+2\\sqrt{10}}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{10+2\\sqrt{10}}{2}" }, { "type": "interim", "title": "Factor $$10+2\\sqrt{10}:{\\quad}2\\left(5+\\sqrt{10}\\right)$$", "input": "10+2\\sqrt{10}", "result": "=\\frac{2\\left(5+\\sqrt{10}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:5+2\\sqrt{10}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(5+\\sqrt{10}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=5+\\sqrt{10}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79/ul7VUbFVJ26S67qVBPqeD0gidxjQIJmWscuYRpiz5oHpYayvDNk2OAeHm77brGCUCWbkwGOY7PqKo3U/JLJVTwfe/oaDMdyG/BkvStAUM/y9DKGIPglJ+qMi9xDu2KpFiYRdLv0u9vmLbIinaRruD0gidxjQIJmWscuYRpiz53sNoh6cotOLBm/ZqdLDCN" } }, { "type": "interim", "title": "$$x=\\frac{-\\left(-10\\right)-2\\sqrt{10}}{2\\cdot\\:1}:{\\quad}5-\\sqrt{10}$$", "input": "\\frac{-\\left(-10\\right)-2\\sqrt{10}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{10-2\\sqrt{10}}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{10-2\\sqrt{10}}{2}" }, { "type": "interim", "title": "Factor $$10-2\\sqrt{10}:{\\quad}2\\left(5-\\sqrt{10}\\right)$$", "input": "10-2\\sqrt{10}", "result": "=\\frac{2\\left(5-\\sqrt{10}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:5-2\\sqrt{10}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(5-\\sqrt{10}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=5-\\sqrt{10}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fFFzGKGpQqH1+arANkx3PuD0gidxjQIJmWscuYRpiz5oHpYayvDNk2OAeHm77brGCUCWbkwGOY7PqKo3U/JLJZirszo6yAciMG4EMzu5MVg/y9DKGIPglJ+qMi9xDu2KwhgAJfkHMRD3Tj5KCavWuuD0gidxjQIJmWscuYRpiz53sNoh6cotOLBm/ZqdLDCN" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=5+\\sqrt{10},\\:x=5-\\sqrt{10}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=5+\\sqrt{10},\\:x=5-\\sqrt{10}" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6p3abg0MC1vxWorffr41OhOccnLtMpNEyjgVL8ewgAEvqUE1xsatCkHtfQYqoIkv8GFc2nNahNWcQmyfomJnFSUgcFTh+Pg7MRZzT4GP+eQeo3oe/oyhMy2+1TQhDBd2f1/RDt6XPLyjiMit25raJ2P31dqKnPu+VUTS8EKz0JPa1a/fEl9nEq1bCVIo1BGARg==" } }, { "type": "step", "primary": "The function domain", "result": "x<5-\\sqrt{10}\\lor\\:5-\\sqrt{10}<x<5+\\sqrt{10}\\lor\\:x>5+\\sqrt{10}" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{1}{x^{2}-10x+15}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }