Algebra II B - Spring - PHS - West, Tori / Unit 7 - Rational And Radical Functions18. Find The Inverse Of The Following Function: $\[ Y = (x + 5)^{\frac{1}{2}} \\]Options: A. \[$ Y = X^2 - 5 \$\] B. \[$ X = Y^2 - 5 \$\] C.

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Introduction to Inverse Functions

In algebra, an inverse function is a function that undoes the action of another function. In other words, if we have a function f(x) and its inverse function f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. Inverse functions are used to solve equations and to find the value of a function at a specific point.

Finding the Inverse of a Function

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

  1. Replace f(x) with y: Replace the function f(x) with y. This will give us the equation y = f(x).
  2. Interchange x and y: Interchange the x and y variables. This will give us the equation x = f(y).
  3. Solve for y: Solve the equation x = f(y) for y. This will give us the inverse function f^(-1)(x).

Finding the Inverse of the Given Function

The given function is y = (x + 5)^(1/2). To find the inverse of this function, we need to follow the steps above.

Step 1: Replace f(x) with y

The given function is y = (x + 5)^(1/2). We can replace f(x) with y to get:

y = (x + 5)^(1/2)

Step 2: Interchange x and y

We can interchange the x and y variables to get:

x = (y + 5)^(1/2)

Step 3: Solve for y

To solve for y, we need to isolate y on one side of the equation. We can do this by squaring both sides of the equation:

x^2 = (y + 5)

Now, we can subtract 5 from both sides of the equation to get:

x^2 - 5 = y

Therefore, the inverse of the given function is y = x^2 - 5.

Conclusion

In this article, we learned how to find the inverse of a function. We followed the steps of replacing f(x) with y, interchanging x and y, and solving for y. We applied these steps to find the inverse of the given function y = (x + 5)^(1/2). The inverse of this function is y = x^2 - 5.

Answer

The correct answer is A. y = x^2 - 5.

Practice Problems

  1. Find the inverse of the function y = (x - 3)^(1/2).
  2. Find the inverse of the function y = (x + 2)^(2).
  3. Find the inverse of the function y = (x - 1)^(3).

Solutions

  1. The inverse of the function y = (x - 3)^(1/2) is y = x^2 + 3.
  2. The inverse of the function y = (x + 2)^(2) is y = sqrt(x - 2).
  3. The inverse of the function y = (x - 1)^(3) is y = cuberoot(x + 1).

References

Q&A: Inverse Functions

Q: What is an inverse function?

A: An inverse function is a function that undoes the action of another function. In other words, if we have a function f(x) and its inverse function f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.

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

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

  1. Replace f(x) with y: Replace the function f(x) with y. This will give us the equation y = f(x).
  2. Interchange x and y: Interchange the x and y variables. This will give us the equation x = f(y).
  3. Solve for y: Solve the equation x = f(y) for y. This will give us the inverse function 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) and its inverse f^(-1)(x) are two different functions that are used to solve equations and to find the value of a function at a specific point.

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 value of x corresponds to a unique value of y. If a function is one-to-one, then it has an inverse.

Q: What are some examples of inverse functions?

A: Some examples of inverse functions include:

  • The inverse of the function f(x) = 2x is f^(-1)(x) = x/2.
  • The inverse of the function f(x) = x^2 is f^(-1)(x) = sqrt(x).
  • The inverse of the function f(x) = (x + 5)^(1/2) is f^(-1)(x) = x^2 - 5.

Q: How do I use inverse functions to solve equations?

A: To use inverse functions to solve equations, you need to follow these steps:

  1. Write the equation in function notation: Write the equation in function notation, with the variable x on the left-hand side and the function f(x) on the right-hand side.
  2. Find the inverse of the function: Find the inverse of the function f(x).
  3. Use the inverse function to solve the equation: Use the inverse function to solve the equation for x.

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

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

  • Not following the steps correctly: Make sure to follow the steps of replacing f(x) with y, interchanging x and y, and solving for y.
  • Not checking if the function is one-to-one: Make sure that the function is one-to-one before finding its inverse.
  • Not checking if the inverse function is correct: Make sure that the inverse function is correct by checking if it satisfies the definition of an inverse function.

Conclusion

In this article, we answered some common questions about inverse functions. We discussed what an inverse function is, how to find the inverse of a function, and how to use inverse functions to solve equations. We also discussed some common mistakes to avoid when finding the inverse of a function.