We are given the function $f(x) = \sqrt{2x + 4}$ and asked to find $f^{-1}(10)$, where $f^{-1}(x)$ is the inverse function of $f(x)$.

Step 1: Express $y = f(x)$

Let $y = f(x) = \sqrt{2x + 4}$.

Step 2: Solve for $x$ in terms of $y$

To find the inverse function, solve for $x$ in terms of $y$: $y = \sqrt{2x + 4}$ Square both sides to eliminate the square root: $y^2 = 2x + 4$ Now, solve for $x$: $2x = y^2 - 4$ $x = \frac{y^2 - 4}{2}$

Thus, the inverse function is: $f^{-1}(y) = \frac{y^2 - 4}{2}$

Step 3: Find $f^{-1}(10)$

Substitute $y = 10$ into the inverse function: $f^{-1}(10) = \frac{10^2 - 4}{2} = \frac{100 - 4}{2} = \frac{96}{2} = 48$

Final Answer:

$f^{-1}(10) = 48$

Would you like a detailed explanation on any step or further clarification?

Here are 5 related questions:

1. How do you verify if a function is invertible?
2. Can all functions have an inverse? If not, why?
3. What is the geometric interpretation of an inverse function?
4. How do you find the domain and range of an inverse function?
5. How does the chain rule apply when differentiating inverse functions?

Tip: When solving for inverse functions, always make sure that the original function is one-to-one (passes the horizontal line test) to ensure an inverse exists.