Find fog and gof, if :
`f(x)=x^(2),g(x)=x+1`

Answer

To find \( f \circ g \) (denoted as \( f(g(x)) \)) and \( g \circ f \) (denoted as \( g(f(x)) \)) for the functions \( f(x) = x^2 \) and \( g(x) = x + 1 \), we will follow these steps:

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

1. Start with the function \( g(x) \):
\[ g(x) = x + 1 \]
2. Substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(x + 1) \]
3. Now, apply the function \( f \):
\[ f(x + 1) = (x + 1)^2 \]
4. Expand \( (x + 1)^2 \):
\[ (x + 1)^2 = x^2 + 2x + 1 \]
5. Therefore, we have:
\[ f(g(x)) = x^2 + 2x + 1 \]

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

1. Start with the function \( f(x) \):
\[ f(x) = x^2 \]
2. Substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(x^2) \]
3. Now, apply the function \( g \):
\[ g(x^2) = x^2 + 1 \]
4. Therefore, we have:
\[ g(f(x)) = x^2 + 1 \]

### Final Answers

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