If f : `RrarrR, f(x)=x^(3)` and g : `RrarrR`, g (x) =`2x^(2)+1`, and R is the set of real number, then find fog (x) and gof (x).

Answer

To solve the problem, we need to find the compositions of the functions \( f \) and \( g \). The functions are defined as follows:

- \( f(x) = x^3 \)
- \( g(x) = 2x^2 + 1 \)

We need to find \( f \circ g(x) \) (denoted as \( fog(x) \)) and \( g \circ f(x) \) (denoted as \( gof(x) \)).

### Step 1: Find \( fog(x) \)

1. **Identify \( g(x) \)**:
\[ g(x) = 2x^2 + 1 \]

2. **Substitute \( g(x) \) into \( f \)**:
\[ fog(x) = f(g(x)) = f(2x^2 + 1) \]

3. **Apply the function \( f \)**:
\[ f(2x^2 + 1) = (2x^2 + 1)^3 \]

4. **Expand \( (2x^2 + 1)^3 \)** using the binomial expansion or the identity \( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \):
- Let \( a = 2x^2 \) and \( b = 1 \):
\[ = (2x^2)^3 + 1^3 + 3(2x^2)(1)(2x^2 + 1) \]
\[ = 8x^6 + 1 + 6x^2(2x^2 + 1) \]
\[ = 8x^6 + 1 + 12x^4 + 6x^2 \]

5. **Combine like terms**:
\[ fog(x) = 8x^6 + 12x^4 + 6x^2 + 1 \]

### Step 2: Find \( gof(x) \)

1. **Identify \( f(x) \)**:
\[ f(x) = x^3 \]

2. **Substitute \( f(x) \) into \( g \)**:
\[ gof(x) = g(f(x)) = g(x^3) \]

3. **Apply the function \( g \)**:
\[ g(x^3) = 2(x^3)^2 + 1 \]
\[ = 2x^6 + 1 \]

### Final Results

- \( fog(x) = 8x^6 + 12x^4 + 6x^2 + 1 \)
- \( gof(x) = 2x^6 + 1 \)