Is the inverse a function?
f(x)=(x+9)3
yes
no
Write the inverse function in the form a3
bx+c+d, where a, b, c, and d are constants. Simplify any fractions.
f–
1(x)=

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.

Is the inverse a function? f(x) = (x + 9)^3 yes no Write the inverse function in the form a^3 / (bx + c) + d, where a, b, c, and d are constants. Simplify any fractions. f^(-1)(x) = 

Learn how to find the inverse of the cubic function f(x) = (x + 9)^3. This detailed solution explores the horizontal line test, properties of cubic functions, and step-by-step methods to write the inverse in the form a^3 / (bx + c) + d. Ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation