To find the inverse of the function $f(x) = \sqrt{2x+6}$, follow these steps:

1. Replace $f(x)$ with $y$:

$y = \sqrt{2x + 6}$

2. Swap $x$ and $y$ (this is the essence of finding the inverse):

$x = \sqrt{2y + 6}$

3. Solve for $y$:

• First, square both sides to remove the square root: $x^2 = 2y + 6$
• Subtract 6 from both sides: $x^2 - 6 = 2y$
• Divide by 2 to isolate $y$: $y = \frac{x^2 - 6}{2}$

4. Rewrite the inverse function:

The inverse function is: $f^{-1}(x) = \frac{x^2 - 6}{2}$

Thus, the inverse of $f(x) = \sqrt{2x + 6}$ is $f^{-1}(x) = \frac{x^2 - 6}{2}$.

Would you like a detailed explanation of any of these steps or have any further questions?

Relative Questions:

1. How do you verify if a function is one-to-one before finding its inverse?
2. What is the domain and range of the inverse function?
3. Can every function have an inverse? Why or why not?
4. What happens if the original function is not one-to-one?
5. How does the graph of a function and its inverse relate to each other?

Tip:

To check if you've correctly found the inverse, compose the original function and its inverse: $f(f^{-1}(x))$ and $f^{-1}(f(x))$ should both return $x$.