If the functions `f:RrarrRandg:RrarrR` be defined by `f(x)=2x+1,g(x)=x^(2)-2`. Find the formulae for g o f and f o g.

Answer

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

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

We need to find \( g(f(x)) \) and \( f(g(x)) \).

### Step 1: Find \( g(f(x)) \)

1. **Substitute \( f(x) \) into \( g(x) \)**:
\[ g(f(x)) = g(2x + 1) \]
2. **Use the definition of \( g(x) \)**:
\[ g(x) = x^2 - 2 \]
Therefore,
\[ g(2x + 1) = (2x + 1)^2 - 2 \]
3. **Expand \( (2x + 1)^2 \)**:
\[ (2x + 1)^2 = 4x^2 + 4x + 1 \]
4. **Subtract 2**:
\[ g(f(x)) = 4x^2 + 4x + 1 - 2 = 4x^2 + 4x - 1 \]

### Step 2: Find \( f(g(x)) \)

1. **Substitute \( g(x) \) into \( f(x) \)**:
\[ f(g(x)) = f(x^2 - 2) \]
2. **Use the definition of \( f(x) \)**:
\[ f(x) = 2x + 1 \]
Therefore,
\[ f(x^2 - 2) = 2(x^2 - 2) + 1 \]
3. **Distribute the 2**:
\[ = 2x^2 - 4 + 1 = 2x^2 - 3 \]

### Final Results

- \( g(f(x)) = 4x^2 + 4x - 1 \)
- \( f(g(x)) = 2x^2 - 3 \)