Questions: Write an equation for the inverse of the given one-to-one function. f(x)=9-(x-8)^(1/3) f^(-1)(x)= -x^3+27x^2-243x+737 (Do not multiply. Do not combine like terms.)

Write an equation for the inverse of the given one-to-one function. f(x)=9-(x-8)^(1/3) f^(-1)(x)= -x^3+27x^2-243x+737 (Do not multiply. Do not combine like terms.)
Transcript text: Write an equation for the inverse of the given one-to-one function. \[ f(x)=9-\sqrt[3]{x-8} \] $f^{-1}(x)=$ $\square$ $-x^{3}+27 x^{2}-243 x+737$ (Do not multiply. Do not combine like terms.)
failed

Solution

failed
failed

Solution Steps

To find the inverse of the given function \( f(x) = 9 - \sqrt[3]{x-8} \), we need to follow these steps:

  1. Replace \( f(x) \) with \( y \).
  2. Swap \( x \) and \( y \) to solve for \( y \) in terms of \( x \).
  3. Isolate \( y \) to express the inverse function \( f^{-1}(x) \).
Step 1: Define the Original Function

Given the function: \[ f(x) = 9 - \sqrt[3]{x - 8} \]

Step 2: Replace \( f(x) \) with \( y \)

\[ y = 9 - \sqrt[3]{x - 8} \]

Step 3: Swap \( x \) and \( y \)

\[ x = 9 - \sqrt[3]{y - 8} \]

Step 4: Isolate \( y \)

To isolate \( y \), we first move the constant term to the other side: \[ \sqrt[3]{y - 8} = 9 - x \]

Next, we cube both sides to remove the cube root: \[ y - 8 = (9 - x)^3 \]

Finally, we solve for \( y \): \[ y = (9 - x)^3 + 8 \]

Final Answer

\[ \boxed{f^{-1}(x) = (9 - x)^3 + 8} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Convert a recipe that serves 6 and calls for 3/4 cup sugar to one that serves 20.
failedAnswered
Find an equation in the form of y=f(x) for the graph sketched below. Use base 2 for any logarithmic or exponential functions.
failedAnswered
To simulate the night sky, we will be using the software Stellarium, which is free and easy to download/install. Go to the following link: https://stellarium.org/ and click on the platform you're using (Windows/Mac/Linux etc.) to install Stellarium. If you want to use the web version of Stellarium, consult the pdf document "How to Navigate in Stellarium-Web" in the sub-module. After you install and open Stellarium, you'll immediately find yourself with a view of the sky as seen from a field at the date time of your class. Stellarium sets the starting location to be Paris, France, so you'll need to change the starting location to NYC. If you move the mouse around the screen, you'll find two menu bars, one on the lower-left hand side of the screen, and another on the bottom of the screen. On the left-hand side menu, scroll over the options until you find the location option. Click on this a pop-up window appears with a world map. A red arrow shows your location as well as your latitude longitude. On the right hand side of the world map is a list of locations in Stellarium. Type New York into the search box a list of locations in New York state will appear in the list. Click on New York (sometimes it reads New York City, NY). Make sure the arrow changes position on the world map. Write down the numbers in the 'Latitude' and 'Longitude' boxes for your location (NYC). To make things easier to see near the horizon, you'll want to switch the landscape to get rid of the houses and trees that show up by default. To do this, move the mouse to the left-hand side menu and select "Sky and Viewing Options" In the window that
failedAnswered
Simplify the following expression completely: (2y^3)^3
failedAnswered
Evaluate the function f(r)=sqrt(r+3)+9 at the given values of the independent variable and simplify. a. f(-3) b. f(61) c. f(x-3)