Math Problem Statement

find inverse of the function f(x)=x^5-3 over 2

Solution

To find the inverse of the function f(x)=x532f(x) = \frac{x^5 - 3}{2}, follow these steps:

Step 1: Express the function in terms of yy

Let y=f(x)=x532y = f(x) = \frac{x^5 - 3}{2}.

So, y=x532y = \frac{x^5 - 3}{2}

Step 2: Solve for xx

Now, we need to solve for xx in terms of yy.

First, multiply both sides by 2 to eliminate the fraction: 2y=x532y = x^5 - 3

Next, add 3 to both sides: x5=2y+3x^5 = 2y + 3

Finally, take the 5th root of both sides: x=2y+35x = \sqrt[5]{2y + 3}

Step 3: Write the inverse function

The inverse of the function is: f1(y)=2y+35f^{-1}(y) = \sqrt[5]{2y + 3}

Step 4: Substitute yy back to xx

If you want to express the inverse as a function of xx, replace yy with xx: f1(x)=2x+35f^{-1}(x) = \sqrt[5]{2x + 3}

Thus, the inverse of f(x)=x532f(x) = \frac{x^5 - 3}{2} is: f1(x)=2x+35f^{-1}(x) = \sqrt[5]{2x + 3}

Would you like more details or have any questions?

Here are 5 related questions to explore:

  1. How do you verify if a function has an inverse?
  2. What are the domain and range of f1(x)f^{-1}(x)?
  3. How do you prove that f(f1(x))=xf(f^{-1}(x)) = x?
  4. What happens if you apply transformations to the function f(x)f(x)?
  5. How do you find the derivative of f1(x)f^{-1}(x)?

Tip: Inverse functions "undo" the operations of the original function.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Inverse Functions
Algebraic Manipulation
Radical Equations

Formulas

f^{-1}(x) = \sqrt[5]{2x + 3}

Theorems

Inverse Function Theorem

Suitable Grade Level

Grades 10-12