domain of (x+9)/(x^2-9)

{ "query": { "display": "domain $$\\frac{x+9}{x^{2}-9}$$", "symbolab_question": "FUNCTION#domain \\frac{x+9}{x^{2}-9}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "x<-3\\lor -3<x<3\\lor x>3", "interval": "(-\\infty ,-3)\\cup (-3,3)\\cup (3,\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\frac{x+9}{x^{2}-9}\\::{\\quad}x<-3\\lor\\:-3<x<3\\lor\\:x>3$$", "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:$${\\quad}x=3,\\:x=-3$$", "input": "\\frac{x+9}{x^{2}-9}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{x+9}{x^{2}-9}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{2}-9=0:{\\quad}x=3,\\:x=-3$$", "input": "x^{2}-9=0", "steps": [ { "type": "interim", "title": "Move $$9\\:$$to the right side", "input": "x^{2}-9=0", "result": "x^{2}=9", "steps": [ { "type": "step", "primary": "Add $$9$$ to both sides", "result": "x^{2}-9+9=0+9" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=9" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7IGfsPE9RU3/JW76ghU+JwGHPhNMD4vIbnfmbOPIaOcHXPmAATNzpJhto1ADoc8UIS8DfmRBUqpCWOuo05WrN0LyqZizgxqQXqQv8DcRaGZRac2OO74kw6Yo+vX25vsHg0hYzF6vxjSZoMSInCY/PPQbO95/2yiZOrh8Vdx/OHPuzjvyW+SoIKNY+UqExWjAZYg1zx05wPYvtn9pf+3D1D8J9e1u2YCYn2ZgVDfYZK5CVTsxzqLIkZd1EulMXeg/jLGOTf471wbbO4db69yfyEsQVTIXxHbnuqrGIe5Uv+ELDIUjprCA+XIptxfb/EbOOB4CES9nGxD2S5ut+HDkiOxtWKyJwU+3om7eHGIbx8A80UCRHZyRKDzQfKBxnNPRTvJXrhRQTKZrV0QqJ1oeAGfve1FkLGYy7pfNHW7ge8PWtSON6otHROlKM8B29ZaSGj9t9gxR2M3dAGoFCeMW355RtZJJOUDSa/tucvyzZL8i/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "x=\\sqrt{9},\\:x=-\\sqrt{9}" }, { "type": "interim", "title": "$$\\sqrt{9}=3$$", "input": "\\sqrt{9}", "steps": [ { "type": "step", "primary": "Factor the number: $$9=3^{2}$$", "result": "=\\sqrt{3^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{3^{2}}=3$$" ], "result": "=3" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74zYoh2RHt5bF5LAil06rQ913jtrSFDx+UNsawjlOjV0Fuo8niSX4SUBqNT4QFNxbcubCnYZOJ5L8/2gsdymw1KIDQ0pFG0qBgUJZZplUxb0=" } }, { "type": "interim", "title": "$$-\\sqrt{9}=-3$$", "input": "-\\sqrt{9}", "steps": [ { "type": "step", "primary": "Factor the number: $$9=3^{2}$$", "result": "=-\\sqrt{3^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{3^{2}}=-3$$" ], "result": "=-3" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l9NogLgG1Z80hBQjGI22siAn9lkDfZkicUGkO3EF+Ir6E5Lqm+d+IsRw+A/8SCikMHIu6EZfZrJ7HpyNTqg74jkiBatsWcqNKMJJIDL8SJQ=" } }, { "type": "step", "result": "x=3,\\:x=-3" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=3,\\:x=-3" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6sy4e6jK4kujI4ULqoVJMyDD8IqnVMcxtixJKRV0/m09gNghAZYORfq9JtKwOORytPfV2oqc+75VRNLwQrPQk9pugt1g9hg3EUIob9qhnfZUgQUxJPyUNnGfVirkcwpVO92jAB5RE7BOdqi256y9oUI/L/NWLiz7WlXYJyi0meZnYoDSQN4diQ31DYw8UMxnuQ==" } }, { "type": "step", "primary": "The function domain", "result": "x<-3\\lor\\:-3<x<3\\lor\\:x>3" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{x+9}{x^{2}-9}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }