{
"query": {
"display": "rango $$f\\left(x\\right)=\\frac{8x}{9x-1}$$",
"symbolab_question": "FUNCTION#range f(x)=\\frac{8x}{9x-1}"
},
"solution": {
"level": "PERFORMED",
"subject": "Functions & Graphing",
"topic": "Functions",
"subTopic": "range",
"default": "f(x)<\\frac{8}{9}\\lor f(x)>\\frac{8}{9}",
"interval": "(-\\infty ,\\frac{8}{9})\\cup (\\frac{8}{9},\\infty )",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "Rango de $$\\frac{8x}{9x-1}:{\\quad}f\\left(x\\right)<\\frac{8}{9}\\lor\\:f\\left(x\\right)>\\frac{8}{9}$$",
"steps": [
{
"type": "definition",
"title": "Definición de rango de función",
"text": "El conjunto de valores de la variable dependiente para la que se define una función"
},
{
"type": "step",
"primary": "El rango de una función es el dominio combinado de las funciones inversas"
},
{
"type": "step",
"primary": "Encontrar encontrar la(s) funcion(es) inversa(s) de: $$\\frac{8x}{9x-1}$$"
},
{
"type": "interim",
"title": "Inversa de $$\\frac{8x}{9x-1}:{\\quad}\\frac{x}{9x-8}$$",
"steps": [
{
"type": "definition",
"title": "Definición de funcion inversa",
"text": "Una función g es la inversa de la función f if for $$y=f\\left(x\\right),\\:\\:x=g\\left(y\\right)\\:$$"
},
{
"type": "step",
"result": "y=\\frac{8x}{9x-1}"
},
{
"type": "interim",
"title": "Reemplazar $$x\\:$$con $$y$$",
"input": "y=\\frac{8x}{9x-1}",
"result": "x=\\frac{8y}{9y-1}",
"steps": [
{
"type": "step",
"primary": "Reemplazar $$x\\:$$con $$y$$",
"secondary": [
"Reemplazar $$y\\:$$con $$x$$"
],
"result": "x=\\frac{8y}{9y-1}"
}
],
"meta": {
"interimType": "Interchange Variables 2Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZZZ/AHPUxGsBjBLoBvG3NA6XLy4cXbcOhDbKXjjQr57G3SdWT3E2VyK1nevy294PYnvEZy18Tqg4s9eiaTh1Kha9dDZkXCs5+Cxg1YNvJNPUbugb/KgY0jmNARlpu4D9WC5O1C7yklGDS3bhKHDYPOY="
}
},
{
"type": "interim",
"title": "Resolver para $$y,\\:x=\\frac{8y}{9y-1}$$",
"input": "x=\\frac{8y}{9y-1}",
"steps": [
{
"type": "interim",
"title": "Multiplicar ambos lados por $$9y-1$$",
"input": "x=\\frac{8y}{9y-1}",
"result": "x\\left(9y-1\\right)=8y",
"steps": [
{
"type": "step",
"primary": "Multiplicar ambos lados por $$9y-1$$",
"result": "x\\left(9y-1\\right)=\\frac{8y}{9y-1}\\left(9y-1\\right)"
},
{
"type": "step",
"primary": "Simplificar",
"result": "x\\left(9y-1\\right)=8y"
}
],
"meta": {
"interimType": "Multiply Both Sides Specific 1Eq",
"gptData": "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"
}
},
{
"type": "interim",
"title": "Desarrollar $$x\\left(9y-1\\right):{\\quad}9xy-x$$",
"input": "x\\left(9y-1\\right)",
"steps": [
{
"type": "step",
"primary": "Poner los parentesis utilizando: $$a\\left(b-c\\right)=ab-ac$$",
"secondary": [
"$$a=x,\\:b=9y,\\:c=1$$"
],
"result": "=x\\cdot\\:9y-x\\cdot\\:1",
"meta": {
"practiceLink": "/practice/expansion-practice",
"practiceTopic": "Expand Rules"
}
},
{
"type": "step",
"result": "=9xy-1\\cdot\\:x"
},
{
"type": "step",
"primary": "Multiplicar: $$1\\cdot\\:x=x$$",
"result": "=9xy-x"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Expand Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75e62/U1fKTRee7GPa/r/c3cUJadsLvGcDY7IUPYXjf6zs903yhxK0NInTwR7JvSRyM/29l7SHlwNleTQoRn0qGbU+JSSzhwT4U9vUdKov1SP48s82QnkM6cIFZ18fbZQYcTfc1m1XpAfMZNyVDp9Wg=="
}
},
{
"type": "step",
"result": "9xy-x=8y"
},
{
"type": "interim",
"title": "Desplace $$x\\:$$a la derecha",
"input": "9xy-x=8y",
"result": "9xy=8y+x",
"steps": [
{
"type": "step",
"primary": "Sumar $$x$$ a ambos lados",
"result": "9xy-x+x=8y+x"
},
{
"type": "step",
"primary": "Simplificar",
"result": "9xy=8y+x"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73EcCVWwkV3f4SjSGwo0YEjigGhg3/px8WfZYMphU3q4IjMsg7ucKQR4ejQgYNe9lctHtnl7uLrSWH3X6GsUMR7NFLEfUPN4Xu/5nS4ys9OCagcfZnFW6jrf88qUT8Cn82P9G1NgmTAdEyme8zwDr9QpcRLepjOMMzCaiiOME7MG+x6pc4ywyIla03xmM7JEYCZk5hyvLR6Ofk1BvX+QuuSBWdZYE75oOP7QUxbjhE5btInqW19JFEhICFKEubrr8U9g9Rz8CeeI5IjmfC1pAYMuTpO5cF+LU0JSZHsAGqIxwJyBPwcQYfezkAXacjWzFAgrmVmO+3SfzGVR1qfJHbX6hIXqtS5CCJpbMhOmQNWGYqgot4ihR+XmeB5qqQo9VN753R8blS5NuKCOQDvP3LM0X8Ckoj3PuQO6J7hSr+nGINaFVrjHZk47W9oenEOqQuYpQ0n6BB/nWQWb8jHV15HkczN78p6sSSiIN47RfWQmPnUcJGp2WiVMMmEnAtUws75VLPbLDeBrnUBvlTj9xp8rfCxUJuVKbG7GeYn05IRRkaSCajGg2VicA6RuUlsHLHe2RukSECEepZtRJJe9Uu0UqTd96MWTKI6Kr2Ib0iQDN6QiUzhQE/VSXNunSX91r5osQXN2tTdIHJPVtIt5cLQRf/E0G5+R2415ctsq82+A="
}
},
{
"type": "interim",
"title": "Desplace $$8y\\:$$a la izquierda",
"input": "9xy=8y+x",
"result": "9xy-8y=x",
"steps": [
{
"type": "step",
"primary": "Restar $$8y$$ de ambos lados",
"result": "9xy-8y=8y+x-8y"
},
{
"type": "step",
"primary": "Simplificar",
"result": "9xy-8y=x"
}
],
"meta": {
"interimType": "Move to the Left Title 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ejYAYJ8nckFdNCJXxM4GlHxDTSYMRPTvSFrCwlaJ8joaQWmymJPDRx9k1PrcazIhsEF6ychL0GXXAwF5wOLOQCAyoJ9JsrXFdpHrCpF+v4aBH1yT/8OkYltIvlibtfgoz27xpiicCLlej+r3Kqpg6rOnDLD0jssdVCDGYRPtoVR0UNymm/R2MkGca/lVQnha5pYT1W5at3qo0hrTO87583U3KYPb3+06je+7A5ci77LPdBWmMWpUWlP8hR3KIpvKFO0lWmuP2kZe8vwVTz2kS4w0UUjML6knBVilv9UfZ4y0Xe8IVlMZzsmEn/FaTl/l0xJ7/3skxGOIhox+LD6iilmu8EugKzlD1uYknZGqg2Wh82wsU1loXzJcOnASu0nhLhEVOZHRag74H4u5Bf4LUHLUQIzcmQ+wDL03Q9pBGhjDu2lvhAiRDtY1vPrrakqZaI1k5/26GQQd5oeT5wk7KqU5YxDLzW4dJ11H1VcrdkO0liQlMRTzJUmAIbvvxAK5bgJv9ltOmEssIBRWD1cpqd8dD0Lo6CIUNEYCqtsfAHgm/U1KHQsVoHy2nFpFSDpBjjVSpBZkaPfgpLmaKJuiP3x+pFb2rvhDUe2sIQFPphL5VxUIidZMoxBuhBhzPiSG6g5JlXSok89je74ErzCB440MB3q0E6IkiSN5ug6MlHYkt3WiGR7ZaCaXvz77bMjS"
}
},
{
"type": "interim",
"title": "Factorizar $$9xy-8y:{\\quad}y\\left(9x-8\\right)$$",
"input": "9xy-8y",
"steps": [
{
"type": "step",
"primary": "Factorizar el termino común $$y$$",
"result": "=y\\left(9x-8\\right)",
"meta": {
"practiceLink": "/practice/factoring-practice",
"practiceTopic": "Factoring"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Factor Title 1Eq"
}
},
{
"type": "step",
"result": "y\\left(9x-8\\right)=x"
},
{
"type": "interim",
"title": "Dividir ambos lados entre $$9x-8$$",
"input": "y\\left(9x-8\\right)=x",
"result": "y=\\frac{x}{9x-8}",
"steps": [
{
"type": "step",
"primary": "Dividir ambos lados entre $$9x-8$$",
"result": "\\frac{y\\left(9x-8\\right)}{9x-8}=\\frac{x}{9x-8}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "y=\\frac{x}{9x-8}"
}
],
"meta": {
"interimType": "Divide Both Sides Specific 1Eq",
"gptData": "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"
}
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Generic Solve For Title 2Eq"
}
},
{
"type": "step",
"result": "\\frac{x}{9x-8}"
}
],
"meta": {
"solvingClass": "Function Inverse",
"interimType": "Function Inverse Top 1Eq"
}
},
{
"type": "step",
"result": "\\frac{x}{9x-8}"
},
{
"type": "step",
"primary": "Encontrar el dominio de cada función inversa"
},
{
"type": "interim",
"title": "Dominio de $$\\frac{x}{9x-8}\\::{\\quad}x<\\frac{8}{9}\\lor\\:x>\\frac{8}{9}$$",
"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):$${\\quad}x=\\frac{8}{9}$$",
"input": "\\frac{x}{9x-8}",
"steps": [
{
"type": "step",
"primary": "Tomar el(los) denominador(es) de $$\\frac{x}{9x-8}$$ y comparar con cero"
},
{
"type": "interim",
"title": "Resolver $$9x-8=0:{\\quad}x=\\frac{8}{9}$$",
"input": "9x-8=0",
"steps": [
{
"type": "interim",
"title": "Desplace $$8\\:$$a la derecha",
"input": "9x-8=0",
"result": "9x=8",
"steps": [
{
"type": "step",
"primary": "Sumar $$8$$ a ambos lados",
"result": "9x-8+8=0+8"
},
{
"type": "step",
"primary": "Simplificar",
"result": "9x=8"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aV1H1GxEaPVva0JqJP93HRyOUtErWNRXbTqqjj69E3uRMYIxk7g3b/ddgKz030reoyVEP58EvQLVNTiw4nE6Fje7I0qpr3b3ybyIR18Rcm+yiHU3VsxZ0Jvlgz1Dh/2W0zIpeqOKDWh7ydUbxgKhdXr3g+Hzr2XIOhBR2yznfmtJQ29i6a1ZHpqmA78rS9tPWPWX/UBxJBp5HcwsJxH2CgEZeBuDysajQrIK4kFiE9ez3AUdbrVlQyqOfgztOiApN1sX9gMSmflfZyY0Xhe3p6YQzr/+5VK0S7UNlretu+zZbWiE2nf2Ysq7tCleECsPCjunfGW6nNPnuVJD0diRkq2zXb9aXkBsxyzpoouV5pgsOiSVJ7YXdTzVsH6Yga49n3N7uY0qhuA7OdwpuE4JIzqH6bywA5RakNDyiaEb4Bp4T9tJV2mcawG0x++B7k/Zg2aet4QRoY7JL7g3632/WfjOO8FMKH3m3HNm+IZQwwl4Hck0iAC3iiRzndG6W/mpLF3/c5BjGayvlHoCvIRals21b7G3thcAeedRd86Gm3YXKU3XYVwej6XZ32oGxL/cfnP0mRiiZT0/VgZMVEWoN03kCh3oevUunZ7/b0qFKBToQWuK+cWpVpbnknFiaIQVN4hPwGtqHmAtffohxOR6T8JTSnixZZURyqOm0H+cDrU="
}
},
{
"type": "interim",
"title": "Dividir ambos lados entre $$9$$",
"input": "9x=8",
"result": "x=\\frac{8}{9}",
"steps": [
{
"type": "step",
"primary": "Dividir ambos lados entre $$9$$",
"result": "\\frac{9x}{9}=\\frac{8}{9}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "x=\\frac{8}{9}"
}
],
"meta": {
"interimType": "Divide Both Sides Specific 1Eq",
"gptData": "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"
}
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Generic Solve Title 1Eq"
}
},
{
"type": "step",
"primary": "Los siguientes puntos no están definidos",
"result": "x=\\frac{8}{9}"
}
],
"meta": {
"interimType": "Undefined Points 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6soR7Fp9tE97oF13SXjPKZWD0yqNoTWLwGuqxO0C0cFV5ApqFyraVAYbrSDuxrlIa8J9e1u2YCYn2ZgVDfYZK5CRcW/agFqpLn6h+exKVoarQMuNfsNxi0HnireB5E16MTqdkVSvcG+iVOmtn/YqPUMlfKGUlbHqElxp678id8RUF3C0aD/NmBw4BAzUHAyt5zAJzpXYgVu2y16nzD1HI8o="
}
},
{
"type": "step",
"primary": "El dominio de la función",
"result": "x<\\frac{8}{9}\\lor\\:x>\\frac{8}{9}"
}
],
"meta": {
"solvingClass": "Function Domain",
"interimType": "Function Domain Top 1Eq"
}
},
{
"type": "step",
"primary": "Combinar los rangos",
"result": "f\\left(x\\right)<\\frac{8}{9}\\lor\\:f\\left(x\\right)>\\frac{8}{9}"
}
],
"meta": {
"solvingClass": "Function Range"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "\\frac{8x}{9x-1}"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solución
rango
Solución
+1
Notación de intervalos
Pasos de solución
El rango de una función es el dominio combinado de las funciones inversas
Encontrar encontrar la(s) funcion(es) inversa(s) de:
Inversa de
Encontrar el dominio de cada función inversa
Dominio de
Combinar los rangos