Select The Correct Answer.Which Function Is The Inverse Of $f(x)=\frac{\sqrt{x-2}}{6}$?A. F − 1 ( X ) = 36 X 2 + 2 F^{-1}(x)=36 X^2+2 F − 1 ( X ) = 36 X 2 + 2 , For X ≥ 0 X \geq 0 X ≥ 0 B. F − 1 ( X ) = 6 X 2 + 2 F^{-1}(x)=6 X^2+2 F − 1 ( X ) = 6 X 2 + 2 , For X ≥ 0 X \geq 0 X ≥ 0 C. F − 1 ( X ) = 36 X + 2 F^{-1}(x)=36 X+2 F − 1 ( X ) = 36 X + 2 , For

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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 input x. Inverse functions are denoted by the symbol f^(-1) and are used to solve equations and find the values of unknown variables.

The Given Function

The given function is f(x) = √(x-2)/6. This function takes an input x, subtracts 2 from it, takes the square root of the result, and then divides it by 6. To find the inverse function, we need to reverse this process.

Finding the Inverse Function

To find the inverse function, we start by writing y = √(x-2)/6. We then swap the x and y variables to get x = √(y-2)/6. Next, we square both sides of the equation to get x^2 = (y-2)/6. We then multiply both sides by 6 to get 6x^2 = y-2. Finally, we add 2 to both sides to get 6x^2 + 2 = y.

The Inverse Function

The inverse function is f^(-1)(x) = 6x^2 + 2. This function takes an input x, squares it, multiplies the result by 6, and then adds 2.

Comparing the Options

Now that we have found the inverse function, we can compare it to the options given in the problem.

  • Option A: f^(-1)(x) = 36x^2 + 2. This option is incorrect because the coefficient of x^2 is 36, not 6.
  • Option B: f^(-1)(x) = 6x^2 + 2. This option is correct because it matches the inverse function we found.
  • Option C: f^(-1)(x) = 36x + 2. This option is incorrect because it does not match the inverse function we found.

Conclusion

In conclusion, the inverse function of f(x) = √(x-2)/6 is f^(-1)(x) = 6x^2 + 2. This function takes an input x, squares it, multiplies the result by 6, and then adds 2.

Key Takeaways

  • Inverse functions are used to reverse the operation of another function.
  • To find the inverse function, we need to swap the x and y variables and then solve for y.
  • The inverse function is denoted by the symbol f^(-1).
  • Inverse functions are used to solve equations and find the values of unknown variables.

Real-World Applications

Inverse functions have many real-world applications, including:

  • 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.
  • Computer Science: Inverse functions are used in algorithms and data structures.

Common Mistakes

  • Not swapping the x and y variables: This is a common mistake when finding the inverse function.
  • Not solving for y: This is another common mistake when finding the inverse function.
  • Not checking the domain and range: This is an important step when finding the inverse function.

Tips and Tricks

  • Use algebraic manipulations: Inverse functions can be found using algebraic manipulations.
  • Use graphical methods: Inverse functions can be found using graphical methods.
  • Use numerical methods: Inverse functions can be found using numerical methods.

Practice Problems

  • Find the inverse function of f(x) = 2x + 1.
  • Find the inverse function of f(x) = x^2 + 1.
  • Find the inverse function of f(x) = √x.

Solutions

  • The inverse function of f(x) = 2x + 1 is f^(-1)(x) = (x-1)/2.
  • The inverse function of f(x) = x^2 + 1 is f^(-1)(x) = √(x-1).
  • The inverse function of f(x) = √x is f^(-1)(x) = x^2.

Conclusion

Q&A: Inverse Functions

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 input x.

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

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

  1. Write the function as y = f(x).
  2. Swap the x and y variables to get x = f(y).
  3. Solve for y to get y = f^(-1)(x).

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 f(x) maps an input x to an output f(x), while the inverse function f^(-1)(x) maps the output f(x) back to the input x.

Q: Why are inverse functions important?

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

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 find the value of the unknown variable.

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.

Q: Can I use inverse functions to find the domain and range of a function?

A: Yes, you can use inverse functions to find the domain and range of a function. By using the inverse function, you can find the values of x that correspond to the given values of y.

Q: What are some common mistakes to avoid when finding the inverse function?

A: Some common mistakes to avoid when finding the inverse function include:

  • Not swapping the x and y variables
  • Not solving for y
  • Not checking the domain and range

Q: How do I check if a function is one-to-one?

A: To check if a function is one-to-one, you need to check if each output value corresponds to exactly one input value. You can do this by graphing the function and checking if it passes the horizontal line test.

Q: Can I use inverse functions to solve systems of equations?

A: Yes, you can use inverse functions to solve systems of equations. By using the inverse function, you can find the values of the unknown variables.

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

A: To find the inverse function of a composite function, you need to follow these steps:

  1. Find the inverse function of each component function.
  2. Use the inverse functions to find the inverse function of the composite function.

Q: Can I use inverse functions to find the derivative of a function?

A: Yes, you can use inverse functions to find the derivative of a function. By using the inverse function, you can find the derivative of the function.

Q: How do I find the inverse function of a function with a square root?

A: To find the inverse function of a function with a square root, you need to follow these steps:

  1. Write the function as y = f(x).
  2. Swap the x and y variables to get x = f(y).
  3. Solve for y to get y = f^(-1)(x).

Q: Can I use inverse functions to solve optimization problems?

A: Yes, you can use inverse functions to solve optimization problems. By using the inverse function, you can find the maximum or minimum value of a function.

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

A: To find the inverse function of a function with a logarithm, you need to follow these steps:

  1. Write the function as y = f(x).
  2. Swap the x and y variables to get x = f(y).
  3. Solve for y to get y = f^(-1)(x).

Q: Can I use inverse functions to solve differential equations?

A: Yes, you can use inverse functions to solve differential equations. By using the inverse function, you can find the solution to the differential equation.

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

A: To find the inverse function of a function with a trigonometric function, you need to follow these steps:

  1. Write the function as y = f(x).
  2. Swap the x and y variables to get x = f(y).
  3. Solve for y to get y = f^(-1)(x).

Q: Can I use inverse functions to solve partial differential equations?

A: Yes, you can use inverse functions to solve partial differential equations. By using the inverse function, you can find the solution to the partial differential equation.

Conclusion

In conclusion, inverse functions are an important concept in mathematics. They are used to reverse the operation of another function and have many real-world applications. By following the steps outlined in this article, you can find the inverse function of any given function.