Verify that your equation is correct by showing that f(f^-1(x)) = x and f^-1(f(x)) = x

$$ 3\sqrt{x-9} $$

To verify that $ f(f^{-1}(x)) = x $ and $ f^{-1}(f(x)) = x $, we need to:

1. Define the function $ f(x) $ and find its inverse $ f^{-1}(x) $.
2. Show that applying $ f $ to $ f^{-1}(x) $ results in $ x $, and applying $ f^{-1} $ to $ f(x) $ results in $ x $.

### Step 1: Define the function $ f(x) $
The given function is:
\[
f(x) = 3\sqrt{x - 9}
\]

### Step 2: Find the inverse $ f^{-1}(x) $
To find the inverse, we need to solve for $ x $ in terms of $ y $, where:
\[
y = 3\sqrt{x - 9}
\]

1. Divide both sides by 3:
\[
\frac{y}{3} = \sqrt{x - 9}
\]

2. Square both sides to eliminate the square root:
\[
\left( \frac{y}{3} \right)^2 = x - 9
\]

3. Solve for $ x $:
\[
x = \left( \frac{y}{3} \right)^2 + 9
\]

Thus, the inverse function is:
\[
f^{-1}(x) = \left( \frac{x}{3} \right)^2 + 9
\]

### Step 3: Verify that $ f(f^{-1}(x)) = x $
Now, let's substitute $ f^{-1}(x) $ into $ f(x) $ and simplify:

\[
f(f^{-1}(x)) = 3\sqrt{\left( \frac{x}{3} \right)^2 + 9 - 9}
\]

Simplifying inside the square root:
\[
f(f^{-1}(x)) = 3\sqrt{\left( \frac{x}{3} \right)^2}
\]
\[
f(f^{-1}(x)) = 3 \cdot \frac{x}{3} = x
\]

So, we have shown that $ f(f^{-1}(x)) = x $.

### Step 4: Verify that $ f^{-1}(f(x)) = x $
Now, let's substitute $ f(x) $ into $ f^{-1}(x) $ and simplify:

\[
f^{-1}(f(x)) = \left( \frac{3\sqrt{x - 9}}{3} \right)^2 + 9
\]

Simplifying:
\[
f^{-1}(f(x)) = (\sqrt{x - 9})^2 + 9
\]
\[
f^{-1}(f(x)) = (x - 9) + 9 = x
\]

So, we have shown that $ f^{-1}(f(x)) = x $.

### Conclusion:
Both conditions are satisfied:
\[
f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x
\]

This verifies that the inverse function is correct.

Would you like more details, or do you have any questions?

Here are 5 related questions to explore:
1. How do you find the domain and range of $ f(x) $ and $ f^{-1}(x) $?
2. What is the graphical relationship between a function and its inverse?
3. How would this process change if the function were cubic instead of square-root-based?
4. Can every function have an inverse? What are the conditions for invertibility?
5. How can you verify that a function is one-to-one before finding its inverse?

**Tip**: To confirm that a function has

Verify that your equation is correct by showing that f(f^-1(x)) = x and f^-1(f(x)) = x for the function f(x) = 3√(x - 9).

This page demonstrates how to verify that f(f^-1(x)) = x and f^-1(f(x)) = x for the function f(x) = 3√(x - 9). Learn about inverse functions, key algebraic techniques, and the square root function. Suitable for high school students in Grades 10-12.

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