Select The Correct Answer.Which Function Is The Inverse Of $f(x) = (x-3)^3 + 2$?A. $f^{-1}(x) = \sqrt[3]{x-2} + 3$ B. $f^{-1}(x) = \sqrt[3]{x+3} - 2$ C. $f^{-1}(x) = \sqrt[3]{x+2} - 3$ D. $f^{-1}(x) =

by ADMIN 204 views

Understanding Inverse Functions

In mathematics, an inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function f^(-1)(x) maps the output f(x) back to the original input x. Inverse functions are denoted by the notation f^(-1)(x) and are used to solve equations and find the values of unknown variables.

The Given Function

The given function is f(x) = (x-3)^3 + 2. To find the inverse function, we need to reverse the operation of this function. This means that we need to isolate the input x in terms of the output f(x).

Step 1: Replace f(x) with y

To start, we replace f(x) with y, which gives us the equation y = (x-3)^3 + 2.

Step 2: Interchange x and y

Next, we interchange x and y, which gives us the equation x = (y-3)^3 + 2.

Step 3: Solve for y

Now, we need to solve for y. To do this, we first subtract 2 from both sides of the equation, which gives us x - 2 = (y-3)^3.

Step 4: Take the cube root

Next, we take the cube root of both sides of the equation, which gives us \sqrt[3]{x-2} = y-3.

Step 5: Add 3 to both sides

Finally, we add 3 to both sides of the equation, which gives us \sqrt[3]{x-2} + 3 = y.

The Inverse Function

Therefore, the inverse function of f(x) = (x-3)^3 + 2 is f^(-1)(x) = \sqrt[3]{x-2} + 3.

Comparing with the Options

Now, let's compare our answer with the options given:

A. f^(-1)(x) = \sqrt[3]{x-2} + 3 B. f^(-1)(x) = \sqrt[3]{x+3} - 2 C. f^(-1)(x) = \sqrt[3]{x+2} - 3 D. f^(-1)(x) = \sqrt[3]{x+2} - 3

Our answer matches option A.

Conclusion

In conclusion, the inverse function of f(x) = (x-3)^3 + 2 is f^(-1)(x) = \sqrt[3]{x-2} + 3. This is the correct answer.

Key Takeaways

  • Inverse functions are used to reverse the operation of another function.
  • To find the inverse function, we need to isolate the input x in terms of the output f(x).
  • We can use algebraic manipulations to solve for y and find the inverse function.
  • The inverse function of f(x) = (x-3)^3 + 2 is f^(-1)(x) = \sqrt[3]{x-2} + 3.
    Inverse Functions: A Q&A Guide =====================================

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function f^(-1)(x) maps the output f(x) back to the original input x.

Q: Why are inverse functions important?

A: Inverse functions are important because they allow us to solve equations and find the values of unknown variables. They are also used in many real-world applications, such as physics, engineering, and economics.

Q: How do I find the inverse function of a given function?

A: To find the inverse function of a given function, we need to follow these steps:

  1. Replace the function with y.
  2. Interchange x and y.
  3. Solve for y.
  4. Take the inverse of the resulting expression.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two different functions that are related to each other. The function maps an input x to an output f(x), while the inverse function maps the output f(x) back to the original input x.

Q: Can a function have more than one inverse?

A: No, a function cannot have more than one inverse. The inverse function is unique and is denoted by the notation f^(-1)(x).

Q: How do I know if a function has an inverse?

A: A function has an inverse if it is one-to-one, meaning that each output value corresponds to exactly one input value. If a function is one-to-one, then it has an inverse.

Q: What is the notation for an inverse function?

A: The notation for an inverse function is f^(-1)(x), where f is the original function and x is the input value.

Q: Can I use inverse functions to solve equations?

A: Yes, you can use inverse functions to solve equations. By using the inverse function, you can isolate the variable and find its value.

Q: What are some common examples of inverse functions?

A: Some common examples of inverse functions include:

  • The inverse of f(x) = 2x is f^(-1)(x) = x/2.
  • The inverse of f(x) = x^2 is f^(-1)(x) = \sqrt{x}.
  • The inverse of f(x) = (x-3)^3 + 2 is f^(-1)(x) = \sqrt[3]{x-2} + 3.

Q: How do I graph an inverse function?

A: To graph an inverse function, you can use the following steps:

  1. Graph the original function.
  2. Reflect the graph of the original function across the line y = x.
  3. The resulting graph is the graph of the inverse function.

Q: What are some real-world applications of inverse functions?

A: Some real-world applications of inverse functions include:

  • Physics: Inverse functions are used to describe the motion of objects under the influence of forces.
  • Engineering: Inverse functions are used to design and optimize systems.
  • Economics: Inverse functions are used to model the behavior of economic systems.

Conclusion

In conclusion, inverse functions are an important concept in mathematics that have many real-world applications. By understanding how to find and use inverse functions, you can solve equations and model complex systems.