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

Answer

To find \( f(g(x)) \) and \( g(f(x)) \) for the functions \( f(x) = |x + 1| \) and \( g(x) = 2x - 1 \), we will follow these steps:

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

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

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

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

### Final Answers

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