{
"query": {
"display": "domínio $$\\frac{1}{x^{2}-10x+15}$$",
"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": "Dominio de $$\\frac{1}{x^{2}-10x+15}\\::{\\quad}x<5-\\sqrt{10}\\lor\\:5-\\sqrt{10}<x<5+\\sqrt{10}\\lor\\:x>5+\\sqrt{10}$$",
"steps": [
{
"type": "definition",
"title": "Definición de dominio",
"text": "El dominio de una función es el conjunto de entradas o valores de los argumentos para los cuales la función es real y definida"
},
{
"type": "interim",
"title": "Encontrar los puntos no definidos (singularidades):$${\\quad}x=5+\\sqrt{10},\\:x=5-\\sqrt{10}$$",
"input": "\\frac{1}{x^{2}-10x+15}",
"steps": [
{
"type": "step",
"primary": "Tomar el(los) denominador(es) de $$\\frac{1}{x^{2}-10x+15}$$ y comparar con cero"
},
{
"type": "interim",
"title": "Resolver $$x^{2}-10x+15=0:{\\quad}x=5+\\sqrt{10},\\:x=5-\\sqrt{10}$$",
"input": "x^{2}-10x+15=0",
"steps": [
{
"type": "interim",
"title": "Resolver con la fórmula general para ecuaciones de segundo grado:",
"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": "Formula general para ecuaciones de segundo grado:",
"text": "Para una ecuación de segundo grado de la forma $$ax^2+bx+c=0$$ las soluciones son <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$"
},
{
"type": "step",
"primary": "Para $${\\quad}a=1,\\:b=-10,\\:c=15$$",
"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",
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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": "Aplicar las leyes de los exponentes: $$\\left(-a\\right)^{n}=a^{n},\\:$$si $$n$$ es par",
"secondary": [
"$$\\left(-10\\right)^{2}=10^{2}$$"
],
"result": "=\\sqrt{10^{2}-4\\cdot\\:1\\cdot\\:15}"
},
{
"type": "step",
"primary": "Multiplicar los numeros: $$4\\cdot\\:1\\cdot\\:15=60$$",
"result": "=\\sqrt{10^{2}-60}"
},
{
"type": "step",
"primary": "$$10^{2}=100$$",
"result": "=\\sqrt{100-60}"
},
{
"type": "step",
"primary": "Restar: $$100-60=40$$",
"result": "=\\sqrt{40}"
},
{
"type": "interim",
"title": "Descomposición en factores primos de $$40:{\\quad}2^{3}\\cdot\\:5$$",
"input": "40",
"result": "=\\sqrt{2^{3}\\cdot\\:5}",
"steps": [
{
"type": "step",
"primary": "$$40\\:$$divida por $$2\\quad\\:40=20\\cdot\\:2$$",
"result": "=2\\cdot\\:20"
},
{
"type": "step",
"primary": "$$20\\:$$divida por $$2\\quad\\:20=10\\cdot\\:2$$",
"result": "=2\\cdot\\:2\\cdot\\:10"
},
{
"type": "step",
"primary": "$$10\\:$$divida por $$2\\quad\\:10=5\\cdot\\:2$$",
"result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:5"
},
{
"type": "step",
"primary": "$$2,\\:5$$ son números primos, por lo tanto, no se pueden factorizar mas",
"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+uo29zEltAtWESiYl1jaImkLgU14aj8GshrjgQGcHUKzXDXvUHFbGYo8oQYzxkEVBHgcr74aUFvsoqscfDSjrOwqM/ylFQ1pqIQ=="
}
},
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$a^{b+c}=a^b\\cdot\\:a^c$$",
"result": "=\\sqrt{2^{2}\\cdot\\:2\\cdot\\:5}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$",
"result": "=\\sqrt{2^{2}}\\sqrt{2\\cdot\\:5}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$\\sqrt[n]{a^n}=a$$",
"secondary": [
"$$\\sqrt{2^{2}}=2$$"
],
"result": "=2\\sqrt{2\\cdot\\:5}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Simplificar",
"result": "=2\\sqrt{10}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xYKyhiwThaKsz2Tl55KkNEFj7CiO8FQNDSt2Nz1mnF7YCxG04c5OKs3H98sf3K9Zq47vuWedXv2WUg94ER8IwfirhMN65g/lcrrrD1WowQqjeh7+jKEzLb7VNCEMF3Z/bMa9W/vMVXpg532eWx0M51MEBIYbd27cthLqV4z2F81K5PcPlwxS+MZ5RLQ+EhleJLd1ohke2Wgml78++2zI0g=="
}
},
{
"type": "step",
"primary": "Separar las soluciones",
"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": "Aplicar la regla $$-\\left(-a\\right)=a$$",
"result": "=\\frac{10+2\\sqrt{10}}{2\\cdot\\:1}"
},
{
"type": "step",
"primary": "Multiplicar los numeros: $$2\\cdot\\:1=2$$",
"result": "=\\frac{10+2\\sqrt{10}}{2}"
},
{
"type": "interim",
"title": "Factorizar $$10+2\\sqrt{10}:{\\quad}2\\left(5+\\sqrt{10}\\right)$$",
"input": "10+2\\sqrt{10}",
"result": "=\\frac{2\\left(5+\\sqrt{10}\\right)}{2}",
"steps": [
{
"type": "step",
"primary": "Reescribir como",
"result": "=2\\cdot\\:5+2\\sqrt{10}"
},
{
"type": "step",
"primary": "Factorizar el termino común $$2$$",
"result": "=2\\left(5+\\sqrt{10}\\right)",
"meta": {
"practiceLink": "/practice/factoring-practice",
"practiceTopic": "Factoring"
}
}
],
"meta": {
"interimType": "Algebraic Manipulation Factor Title 1Eq"
}
},
{
"type": "step",
"primary": "Dividir: $$\\frac{2}{2}=1$$",
"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": "Aplicar la regla $$-\\left(-a\\right)=a$$",
"result": "=\\frac{10-2\\sqrt{10}}{2\\cdot\\:1}"
},
{
"type": "step",
"primary": "Multiplicar los numeros: $$2\\cdot\\:1=2$$",
"result": "=\\frac{10-2\\sqrt{10}}{2}"
},
{
"type": "interim",
"title": "Factorizar $$10-2\\sqrt{10}:{\\quad}2\\left(5-\\sqrt{10}\\right)$$",
"input": "10-2\\sqrt{10}",
"result": "=\\frac{2\\left(5-\\sqrt{10}\\right)}{2}",
"steps": [
{
"type": "step",
"primary": "Reescribir como",
"result": "=2\\cdot\\:5-2\\sqrt{10}"
},
{
"type": "step",
"primary": "Factorizar el termino común $$2$$",
"result": "=2\\left(5-\\sqrt{10}\\right)",
"meta": {
"practiceLink": "/practice/factoring-practice",
"practiceTopic": "Factoring"
}
}
],
"meta": {
"interimType": "Algebraic Manipulation Factor Title 1Eq"
}
},
{
"type": "step",
"primary": "Dividir: $$\\frac{2}{2}=1$$",
"result": "=5-\\sqrt{10}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fFFzGKGpQqH1+arANkx3PuD0gidxjQIJmWscuYRpiz5oHpYayvDNk2OAeHm77brGCUCWbkwGOY7PqKo3U/JLJZirszo6yAciMG4EMzu5MVg/y9DKGIPglJ+qMi9xDu2KwhgAJfkHMRD3Tj5KCavWuuD0gidxjQIJmWscuYRpiz53sNoh6cotOLBm/ZqdLDCN"
}
},
{
"type": "step",
"primary": "Las soluciones a la ecuación de segundo grado son: ",
"result": "x=5+\\sqrt{10},\\:x=5-\\sqrt{10}"
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Generic Solve Title 1Eq"
}
},
{
"type": "step",
"primary": "Los siguientes puntos no están definidos",
"result": "x=5+\\sqrt{10},\\:x=5-\\sqrt{10}"
}
],
"meta": {
"interimType": "Undefined Points 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6p3abg0MC1vxWorffr41OhOccnLtMpNEyjgVL8ewgAEvqUE1xsatCkHtfQYqoIkv8GFc2nNahNWcQmyfomJnFSUgcFTh+Pg7MRZzT4GP+eQeo3oe/oyhMy2+1TQhDBd2f6YQzr/+5VK0S7UNlretu+x6dtcj6LYjH/BYDeyO3mMWNryIZjUBFafODd/0omTen3zTY5d9Et/+0qCGla07FXc="
}
},
{
"type": "step",
"primary": "El dominio de la función",
"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
}
}
Solución
domínio
Solución
+1
Notación de intervalos
Pasos de solución
Encontrar los puntos no definidos (singularidades):
El dominio de la función