domain of (x+3)/(x^2+4x-5)

{ "query": { "display": "domain $$\\frac{x+3}{x^{2}+4x-5}$$", "symbolab_question": "FUNCTION#domain \\frac{x+3}{x^{2}+4x-5}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "x<-5\\lor -5<x<1\\lor x>1", "interval": "(-\\infty ,-5)\\cup (-5,1)\\cup (1,\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\frac{x+3}{x^{2}+4x-5}\\::{\\quad}x<-5\\lor\\:-5<x<1\\lor\\:x>1$$", "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:$${\\quad}x=1,\\:x=-5$$", "input": "\\frac{x+3}{x^{2}+4x-5}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{x+3}{x^{2}+4x-5}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{2}+4x-5=0:{\\quad}x=1,\\:x=-5$$", "input": "x^{2}+4x-5=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "x^{2}+4x-5=0", "result": "{x}_{1,\\:2}=\\frac{-4\\pm\\:\\sqrt{4^{2}-4\\cdot\\:1\\cdot\\:\\left(-5\\right)}}{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 $${\\quad}a=1,\\:b=4,\\:c=-5$$", "result": "{x}_{1,\\:2}=\\frac{-4\\pm\\:\\sqrt{4^{2}-4\\cdot\\:1\\cdot\\:\\left(-5\\right)}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{4^{2}-4\\cdot\\:1\\cdot\\:\\left(-5\\right)}=6$$", "input": "\\sqrt{4^{2}-4\\cdot\\:1\\cdot\\:\\left(-5\\right)}", "result": "{x}_{1,\\:2}=\\frac{-4\\pm\\:6}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{4^{2}+4\\cdot\\:1\\cdot\\:5}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:5=20$$", "result": "=\\sqrt{4^{2}+20}" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=\\sqrt{16+20}" }, { "type": "step", "primary": "Add the numbers: $$16+20=36$$", "result": "=\\sqrt{36}" }, { "type": "step", "primary": "Factor the number: $$36=6^{2}$$", "result": "=\\sqrt{6^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{6^{2}}=6$$" ], "result": "=6", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74jz8pBgj569lwUyDg0Z0NjTF392od8JjfT80CJ7FSjl9mgUBpIERD+LTv/HGlEAZzMFYmi1F5Hg/ibpEToVnY2jymx1aA7ki4iAlqNx++j0oBoNzMgTR9b6rSqzOkA5HNMXf3ah3wmN9PzQInsVKOYFPS5CEo0JHthWv9Hk7WlY=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{x}_{1}=\\frac{-4+6}{2\\cdot\\:1},\\:{x}_{2}=\\frac{-4-6}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$x=\\frac{-4+6}{2\\cdot\\:1}:{\\quad}1$$", "input": "\\frac{-4+6}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-4+6=2$$", "result": "=\\frac{2}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{2}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/rI2zLkfT+6w2IF8U43Q1SJzNSds22HF6C2iiwwKPXMDnzlbPZjyKgy1eUCFsLd5l1K1r0aaVT/E0zZakID9jPAKCfFHLER2iRFCuUUu6VMzbaMyyDOgzNIq+8aKW+XYsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "$$x=\\frac{-4-6}{2\\cdot\\:1}:{\\quad}-5$$", "input": "\\frac{-4-6}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-4-6=-10$$", "result": "=\\frac{-10}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-10}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{10}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{10}{2}=5$$", "result": "=-5" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UCbmMRtPG6PA7f5/hMm7GSJzNSds22HF6C2iiwwKPXMDnzlbPZjyKgy1eUCFsLd5lLqs5cK+xyypYlr41TQQ575EbOPmFIVDZCKr5u8wDXodm8SFSftcwCjLZktMhhLeialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=1,\\:x=-5" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=1,\\:x=-5" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6v5Gc8UFtLZ3SfpNSZECMe9UTbRefOqyowMfp9SaEwaKqUE1xsatCkHtfQYqoIkv8GFc2nNahNWcQmyfomJnFSUgcFTh+Pg7MRZzT4GP+eQeo3oe/oyhMy2+1TQhDBd2f1/RDt6XPLyjiMit25raJ2P31dqKnPu+VUTS8EKz0JPa1a/fEl9nEq1bCVIo1BGARg==" } }, { "type": "step", "primary": "The function domain", "result": "x<-5\\lor\\:-5<x<1\\lor\\:x>1" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{x+3}{x^{2}+4x-5}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }