{
"query": {
"display": "inversa $$f\\left(x\\right)=\\sqrt{3+7x}$$",
"symbolab_question": "FUNCTION#inverse f(x)=\\sqrt{3+7x}"
},
"solution": {
"level": "PERFORMED",
"subject": "Functions & Graphing",
"topic": "Functions",
"subTopic": "inverse",
"default": "\\frac{x^{2}-3}{7}",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "Inversa de $$\\sqrt{3+7x}:{\\quad}\\frac{x^{2}-3}{7}$$",
"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=\\sqrt{3+7x}"
},
{
"type": "interim",
"title": "Reemplazar $$x\\:$$con $$y$$",
"input": "y=\\sqrt{3+7x}",
"result": "x=\\sqrt{3+7y}",
"steps": [
{
"type": "step",
"primary": "Reemplazar $$x\\:$$con $$y$$",
"secondary": [
"Reemplazar $$y\\:$$con $$x$$"
],
"result": "x=\\sqrt{3+7y}"
}
],
"meta": {
"interimType": "Interchange Variables 2Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZeQumNCAZ/5ZQRoJCS6RJdmD0yqNoTWLwGuqxO0C0cFVz5Lb2DzoR+2lK1ma7A4yFw7HH89RUTaeN3FUkEAdGm9x5qY7oT4QfPd1Ddwv4qJzt0mJce+l2JTUln16cOoWS6lqky4uXqcIuOUwtCqPpLc="
}
},
{
"type": "interim",
"title": "Resolver para $$y,\\:x=\\sqrt{3+7y}$$",
"input": "x=\\sqrt{3+7y}",
"steps": [
{
"type": "interim",
"title": "Elevar al cuadrado ambos lados:$${\\quad}x^{2}=3+7y$$",
"input": "x=\\sqrt{3+7y}",
"result": "x^{2}=3+7y",
"steps": [
{
"type": "step",
"result": "x^{2}=\\left(\\sqrt{3+7y}\\right)^{2}"
},
{
"type": "interim",
"title": "Desarrollar $$\\left(\\sqrt{3+7y}\\right)^{2}:{\\quad}3+7y$$",
"input": "\\left(\\sqrt{3+7y}\\right)^{2}",
"steps": [
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$\\sqrt{a}=a^{\\frac{1}{2}}$$",
"result": "=\\left(\\left(3+7y\\right)^{\\frac{1}{2}}\\right)^{2}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$\\left(a^{b}\\right)^{c}=a^{bc}$$",
"result": "=\\left(3+7y\\right)^{\\frac{1}{2}\\cdot\\:2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$\\frac{1}{2}\\cdot\\:2=1$$",
"input": "\\frac{1}{2}\\cdot\\:2",
"result": "=3+7y",
"steps": [
{
"type": "step",
"primary": "Multiplicar fracciones: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{1\\cdot\\:2}{2}"
},
{
"type": "step",
"primary": "Eliminar los terminos comunes: $$2$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Expand Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vz0OwVmz7FbkxdUH+eNAiVaHHFeGsVlSba/pxL/x6RHptRYmj6WUpgaKrNIresW2wNmnTNPfSXaFwWcdWyIh039H/SFX0n2Ge344m0fgESR6L+JmEubFqEnyJ6OQ6zfpeh02/6zn4Kp3zgaHVhrMsYufIAFk8RziPtgOSd4Aptk="
}
},
{
"type": "step",
"result": "x^{2}=3+7y"
}
],
"meta": {
"interimType": "Radicals Square Both Sides Title 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjCBKdOJvtfezbFI4zIaDphv1Sios1Te+I0yK/nO5j0RsmVBAimTWz5/bcyAzDjmFl0bkDgOhbeCghIkNrATUfx+A2dhZsR0jUVxLBB9f34/aUsWN3dQQm05XtKAtxnQaOWkt9WuQZVU+RpoHiHFohy/6RNElaEnyILXdktFL1GD0Q=="
}
},
{
"type": "interim",
"title": "Resolver $$x^{2}=3+7y:{\\quad}y=\\frac{x^{2}-3}{7}$$",
"input": "x^{2}=3+7y",
"steps": [
{
"type": "step",
"primary": "Intercambiar lados",
"result": "3+7y=x^{2}"
},
{
"type": "interim",
"title": "Desplace $$3\\:$$a la derecha",
"input": "3+7y=x^{2}",
"result": "7y=x^{2}-3",
"steps": [
{
"type": "step",
"primary": "Restar $$3$$ de ambos lados",
"result": "3+7y-3=x^{2}-3"
},
{
"type": "step",
"primary": "Simplificar",
"result": "7y=x^{2}-3"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pNdx8selM7GPM1voACEdpUVrx3Ozw4IEnEQys4DnS7MSlpMUa9xiCMZVa3yd1JnBm8TC9vlgXtjrQ7UXLN79HZHkMKOXk9G+xwh0Bv3ZVYSIUr133bL4qahTkKAuipGGU9ik5WINGrsWotLzS7/BjMilG2DUqMrP9Xo55x7KtrASjYkZQN7u4cUDW1dKm5q29p058Px7HARZpBet1gbCRJzD2R1QKlJ45gZn1eICTWrcvrwjyf/cA3NQbbG+4JXJUS8X9ng6mUi32IxbEU5kwafpZTvJtSVLB7MWEsq+Qo1W+6pShiQqYErp9JHZw2WCR+p9PcdLGV2Y23qtRx+/JILKsA73MxANnKMSJYzuojkc0tqYRLKJeQRU1TgF7fJeVfes/snfzU4It1/sLtZZEO9sGZu5A1MXROmEpnxG69qF/gBP2lEoU6gK4Wgq1pu2kZVr3CE20Y8AXkbdT5eHwUdLijWmhOEmWvjLon9gR8w="
}
},
{
"type": "interim",
"title": "Dividir ambos lados entre $$7$$",
"input": "7y=x^{2}-3",
"result": "y=\\frac{x^{2}-3}{7}",
"steps": [
{
"type": "step",
"primary": "Dividir ambos lados entre $$7$$",
"result": "\\frac{7y}{7}=\\frac{x^{2}}{7}-\\frac{3}{7}"
},
{
"type": "interim",
"title": "Simplificar",
"input": "\\frac{7y}{7}=\\frac{x^{2}}{7}-\\frac{3}{7}",
"result": "y=\\frac{x^{2}-3}{7}",
"steps": [
{
"type": "interim",
"title": "Simplificar $$\\frac{7y}{7}:{\\quad}y$$",
"input": "\\frac{7y}{7}",
"steps": [
{
"type": "step",
"primary": "Dividir: $$\\frac{7}{7}=1$$",
"result": "=y"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bDSUqLCDxNBkTY388RFpjC061ljBSPJeENOw2efoSWu6Rbu0VBfiSYvGZCQ6uC9JRSpN33oxZMojoqvYhvSJAJIIjMnWuPDKLVQSmTzSSG5x/u9Gf2mEN5NbMjU5SglusIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "interim",
"title": "Simplificar $$\\frac{x^{2}}{7}-\\frac{3}{7}:{\\quad}\\frac{x^{2}-3}{7}$$",
"input": "\\frac{x^{2}}{7}-\\frac{3}{7}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$",
"result": "=\\frac{x^{2}-3}{7}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HBMy1k3euUkMrxh46KjL3uKaMifbDtyxSiaTnv+pm1ktOtZYwUjyXhDTsNnn6ElrSGpx9e3Lolh3WlYiBDdmOIDiqV9LiwjmI5GzQZHNv+ujeh7+jKEzLb7VNCEMF3Z/mm5TP4WMAPdM1az402agYqMB/mTasBJceuDqvpmzdLj7IqRAv80k8+JxwwioIXvUialcV/dI5TH4fXyp+ncwuA=="
}
},
{
"type": "step",
"result": "y=\\frac{x^{2}-3}{7}"
}
],
"meta": {
"interimType": "Generic Simplify 0Eq"
}
}
],
"meta": {
"interimType": "Divide Both Sides Specific 1Eq",
"gptData": "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"
}
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Generic Solve Title 1Eq"
}
},
{
"type": "step",
"result": "y=\\frac{x^{2}-3}{7}"
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Generic Solve For Title 2Eq"
}
},
{
"type": "step",
"result": "\\frac{x^{2}-3}{7}"
}
],
"meta": {
"solvingClass": "Function Inverse"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "\\sqrt{3+7x}"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solución
inversa
Solución
Pasos de solución
Reemplazar con
Resolver para