10th Grade

{
"voice_prompt": "",
"manuscript": {
"title": {
"text": "How can you find the inverse of a function?",
"audio": "How can you find the inverse of a function?"
},
"description": {
"text": "You can find the inverse of a function by following four steps:\r\n\r\n 1.Replace f of x with y\r\n 2.Swap the roles of x and y\r\n 3. Solve for y\r\n 4. Rewrite using inverse notation",
"audio": "You can find the inverse of a function by following four steps:\r\n\r\n 1.Replace f of x with y\r\n 2.Swap the roles of x and y\r\n 3. Solve for y\r\n 4. Rewrite using inverse notation"
},
"scenes": [
{
"text": "Let's look at an example. Suppose you\u2019re given the function:\r\n f of x equals 2 x plus 3,\r\n and you want to find its inverse.\r\n",
"latex": "f(x) = 2x + 3"
},
{
"text": "You start by rewriting the function using y instead of f of x.\r\nThis gives you:\r\n y equals 2 x plus 3\r\n\r\nYou can do this because f of x and y represent the same thing \u2014 they both show the output of the function. Using y just makes the equation easier to work with.",
"latex": "y = 2x + 3"
},
{
"text": "Next, you switch the roles of x and y.\r\n Why? Because finding the inverse of a function means reversing the inputs and outputs.\r\n\r\nSo, you swap them:\r\nx equals 2 y plus 3",
"latex": "x = 2y + 3"
},
{
"text": "Now you solve this equation for y.\r\n Start by subtracting 3 from both sides:\r\n\r\n x minus 3 equals 2 y",
"latex": "x - 3 = 2y"
},
{
"text": "Next, divide both sides by 2 to isolate y:\r\n\r\n y equals (x minus 3) divided by 2",
"latex": "y = \\frac{x - 3}{2}"
},
{
"text": "Now that you have solved for y, you rewrite it using inverse function notation.\r\nThe inverse of f of x is written as f inverse of x \u2014 that\u2019s f with a little negative one, like this: f to the negative one of x \r\n\r\n So the inverse function is:\r\n\r\n f inverse of x equals (x minus 3) divided by 2",
"latex": "f^{-1}(x) = \\frac{x - 3}{2}"
}
],
"outro": {
"text": "You can find the inverse of a function by following four steps:\r\n\r\n 1.Replace f of x with y\r\n 2.Swap the roles of x and y\r\n 3. Solve for y\r\n 4. Rewrite using inverse notation",
"audio": "You can find the inverse of a function by following four steps:\r\n\r\n 1.Replace f of x with y\r\n 2.Swap the roles of x and y\r\n 3. Solve for y\r\n 4. Rewrite using inverse notation"
}
}
}