To determine if the inverse of the function $f(x) = (x + 9)^3$ is a function and to find the inverse function in the specified form, follow these steps:

Step 1: Check if the Inverse is a Function

The function given is $f(x) = (x + 9)^3$, which is a cubic function. Cubic functions are one-to-one because they pass the horizontal line test (no horizontal line intersects the graph more than once). Therefore, the inverse of this function does exist and is also a function.

Step 2: Find the Inverse Function

To find the inverse, we need to swap $x$ and $y$, and solve for $y$.

1. Start by writing $f(x)$ as $y = (x + 9)^3$. $y = (x + 9)^3$

2. Swap $x$ and $y$. $x = (y + 9)^3$

3. Solve for $y$ by taking the cube root of both sides. $\sqrt[3]{x} = y + 9$

4. Isolate $y$. $y = \sqrt[3]{x} - 9$

Step 3: Write in the Desired Form

The inverse function is: $f^{-1}(x) = \sqrt[3]{x} - 9$ This matches the form $a \cdot \sqrt[3]{x} + b$, where $a = 1$, $b = -9$, and there are no other constants $c$ or $d$ since this is a simple cubic root and subtraction.

Final Answer:

$f^{-1}(x) = \sqrt[3]{x} - 9$

Would you like further details on how to verify this inverse function?

Related Questions:

1. What is the domain and range of the inverse function $f^{-1}(x)$?
2. Can you describe the process for finding the inverse of quadratic functions?
3. How can you graph both $f(x) = (x + 9)^3$ and its inverse on the same coordinate plane?
4. How do transformations like translations affect the inverse of a function?
5. What are some examples of functions that do not have an inverse?

Tip:

Always check the original function’s domain and range to ensure the inverse function applies correctly to the entire domain.