domain of f(x)=(x-5)/(x^2-25)

{ "query": { "display": "domain $$f\\left(x\\right)=\\frac{x-5}{x^{2}-25}$$", "symbolab_question": "FUNCTION#domain f(x)=\\frac{x-5}{x^{2}-25}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "x<-5\\lor -5<x<5\\lor x>5", "interval": "(-\\infty ,-5)\\cup (-5,5)\\cup (5,\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\frac{x-5}{x^{2}-25}\\::{\\quad}x<-5\\lor\\:-5<x<5\\lor\\:x>5$$", "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:$${\\quad}x=5,\\:x=-5$$", "input": "\\frac{x-5}{x^{2}-25}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{x-5}{x^{2}-25}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{2}-25=0:{\\quad}x=5,\\:x=-5$$", "input": "x^{2}-25=0", "steps": [ { "type": "interim", "title": "Move $$25\\:$$to the right side", "input": "x^{2}-25=0", "result": "x^{2}=25", "steps": [ { "type": "step", "primary": "Add $$25$$ to both sides", "result": "x^{2}-25+25=0+25" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=25" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "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{25},\\:x=-\\sqrt{25}" }, { "type": "interim", "title": "$$\\sqrt{25}=5$$", "input": "\\sqrt{25}", "steps": [ { "type": "step", "primary": "Factor the number: $$25=5^{2}$$", "result": "=\\sqrt{5^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{5^{2}}=5$$" ], "result": "=5" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7puAS6/i3sXvnGtmBv+LV0iAn9lkDfZkicUGkO3EF+Ir6qe9KeopEQ7bshfzgNOinzGQBGgzjhY9wy62Ypaxu7OMA6oWp15qVTjRqxzWUFFs=" } }, { "type": "interim", "title": "$$-\\sqrt{25}=-5$$", "input": "-\\sqrt{25}", "steps": [ { "type": "step", "primary": "Factor the number: $$25=5^{2}$$", "result": "=-\\sqrt{5^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{5^{2}}=-5$$" ], "result": "=-5" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pF7MBqwVlZH6D5wE166ynt6GQqufR6tr2vPxOUv7H+/uE+sSwSGkn514HAkT777UrAJFSr9Yxv/LaHr9YfwzxhNYDn5fKk80QYv8lVHzC88=" } }, { "type": "step", "result": "x=5,\\:x=-5" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=5,\\:x=-5" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6pu1yvSpbdS8FHZfhp4yG7ZgYjjCFvIZi1+JfL9rYHG5O/MzuqrtLw992Ul6dK1xH6yibDDkIm6wskon/zIJhI+rhuyXnKnluPLFfVczi25x72wZm7kDUxdE6YSmfEbr2hafHanhoOWpqNiGyp41opN7e4htjydF5+TQXMB3bo9lnKu6BEuIeTBMxtjMl/QpxQ==" } }, { "type": "step", "primary": "The function domain", "result": "x<-5\\lor\\:-5<x<5\\lor\\:x>5" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{x-5}{x^{2}-25}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }