give the inverse function of the function y=\sqrt{2x}

To find the inverse of the function \( y = \sqrt{2x} \), follow these steps:

1. **Express the function in terms of \( y \)**:
   \[
   y = \sqrt{2x}
   \]

2. **Switch \( x \) and \( y \)**:
   To find the inverse, swap \( x \) and \( y \).
   \[
   x = \sqrt{2y}
   \]

3. **Solve for \( y \)**:
   Square both sides to eliminate the square root.
   \[
   x^2 = 2y
   \]
   Divide both sides by 2:
   \[
   y = \frac{x^2}{2}
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{x^2}{2}
\]

Would you like further details or have any questions?

### Related Questions:
1. How can you verify if two functions are inverses of each other?
2. What are the restrictions on the domain and range for inverse functions?
3. How do you find the inverse of more complex functions involving exponents and roots?
4. What is the graphical interpretation of inverse functions?
5. What are the properties of the square root function and how do they affect the inverse?

**Tip**: Always check the domain and range when finding an inverse function, as the original function must be one-to-one.

Discover how to find the inverse function of y=√2x with a clear step-by-step guide. This problem explores inverse functions and radical equations, suitable for high school students in Grades 10-12.

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