If `f(x)=(a x^2+b)^3,` then find the function `g` such that `f(g(x))=g(f(x))dot`

Options

A. `g(x)=((b-x^(1//3))/(a))^(1//2)`
B. `g(x)=(1)/((ax^(2)+b)^(3))`
C. `g(x)=(ax^(2)+b)^(1//3)`
D. `g(x)=((x^(1//3)-b)/(a))^(1//2)`

Answer

To solve the problem, we need to find the function \( g(x) \) such that \( f(g(x)) = g(f(x)) \), where \( f(x) = (a x^2 + b)^3 \).

### Step 1: Understand the function \( f(x) \)
We start with the function:
\[ f(x) = (a x^2 + b)^3 \]

### Step 2: Set up the equation
We need to find \( g(x) \) such that:
\[ f(g(x)) = g(f(x)) \]

### Step 3: Assume \( f(g(x)) \)
Substituting \( g(x) \) into \( f \):
\[ f(g(x)) = (a (g(x))^2 + b)^3 \]

### Step 4: Assume \( g(f(x)) \)
Now substituting \( f(x) \) into \( g \):
\[ g(f(x)) = g((a x^2 + b)^3) \]

### Step 5: Find the inverse function \( f^{-1}(x) \)
To find \( g(x) \), we can assume that \( g(x) = f^{-1}(x) \). Thus, we need to find the inverse of \( f(x) \).

1. Let \( y = f(x) \):
\[ y = (a x^2 + b)^3 \]

2. Taking the cube root on both sides:
\[ y^{1/3} = a x^2 + b \]

3. Rearranging gives:
\[ a x^2 = y^{1/3} - b \]

4. Dividing by \( a \):
\[ x^2 = \frac{y^{1/3} - b}{a} \]

5. Taking the square root:
\[ x = \sqrt{\frac{y^{1/3} - b}{a}} \]

Thus, the inverse function is:
\[ f^{-1}(x) = \sqrt{\frac{x^{1/3} - b}{a}} \]

### Step 6: Define \( g(x) \)
Since we have found \( f^{-1}(x) \), we can define:
\[ g(x) = f^{-1}(x) = \sqrt{\frac{x^{1/3} - b}{a}} \]

### Final Answer
The function \( g(x) \) is:
\[ g(x) = \sqrt{\frac{x^{1/3} - b}{a}} \]