Sure, let's solve the given assignment step by step.
First, let's define our functions:
1. [tex]\( g(x) = x + 1 \)[/tex]
2. [tex]\( f(x) = 3x - 2 \)[/tex]
We need to find [tex]\( g \circ g \)[/tex], which means [tex]\( g(g(x)) \)[/tex].
### Step 1: Calculate [tex]\( g(g(x)) \)[/tex]
Let's first substitute [tex]\( g(x) = x + 1 \)[/tex] back into itself:
[tex]\[ g(g(x)) = g(x + 1) \][/tex]
Now, substitute [tex]\( x + 1 \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(x + 1) = (x + 1) + 1 \][/tex]
[tex]\[ g(x + 1) = x + 2 \][/tex]
So, we have:
[tex]\[ g(g(x)) = x + 2 \][/tex]
### Step 2: Calculate [tex]\( f(g(g(x))) \)[/tex]
Now, we need to substitute [tex]\( g(g(x)) \)[/tex] into [tex]\( f(x) \)[/tex].
[tex]\[ f(g(g(x))) = f(x + 2) \][/tex]
Using the definition of [tex]\( f(x) = 3x - 2 \)[/tex], substitute [tex]\( x + 2 \)[/tex] into [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex]:
[tex]\[ f(x + 2) = 3(x + 2) - 2 \][/tex]
[tex]\[ f(x + 2) = 3x + 6 - 2 \][/tex]
[tex]\[ f(x + 2) = 3x + 4 \][/tex]
Therefore, we have:
[tex]\[ f(g(g(x))) = 3x + 4 \][/tex]
### Summary
- [tex]\( g(x) = x + 1 \)[/tex]
- [tex]\( g(g(x)) = x + 2 \)[/tex]
- [tex]\( f(g(g(x))) = 3x + 4 \)[/tex]
Final results:
[tex]\[
\boxed{g(x) = x + 1, \quad g(g(x)) = x + 2, \quad f(g(g(x))) = 3x + 4}
\][/tex]
