To find the inverse of the function \( f(x) = 9\sqrt{x+8} \), we follow these steps:

### Step 1: Write \( f(x) = y \)
\[
y = 9\sqrt{x+8}
\]

### Step 2: Solve for \( x \) in terms of \( y \)
First, divide both sides by 9:
\[
\frac{y}{9} = \sqrt{x+8}
\]

Now, square both sides to eliminate the square root:
\[
\left(\frac{y}{9}\right)^2 = x + 8
\]

### Step 3: Isolate \( x \)
Subtract 8 from both sides:
\[
x = \left(\frac{y}{9}\right)^2 - 8
\]

### Step 4: Write the inverse function
Now, swap \( x \) and \( y \) to express the inverse function:
\[
f^{-1}(x) = \left(\frac{x}{9}\right)^2 - 8
\]

Thus, the inverse of \( f(x) = 9\sqrt{x+8} \) is:
\[
f^{-1}(x) = \left(\frac{x}{9}\right)^2 - 8
\]

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. What is the domain of the original function \( f(x) = 9\sqrt{x+8} \)?
2. What is the domain and range of the inverse function \( f^{-1}(x) \)?
3. How do you verify that two functions are inverses of each other?
4. How do transformations of the square root function affect its graph?
5. How do you solve equations involving square roots?

**Tip:** When solving for an inverse, remember to restrict the domain if necessary to ensure the inverse is a valid function.

Learn how to find the inverse of the function f(x) = 9√(x+8) with a detailed step-by-step solution. This problem involves algebraic techniques for solving inverse functions and manipulating square roots. Ideal for high school students in Grades 10-12.

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