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Example:
[tex]\[
\begin{aligned}
f(g) & = 3x - 2 \\
& = 3(x + 1) - 2 \\
& = 3x + 3 - 2 \\
& = 3x + 1
\end{aligned}
\][/tex]

Assignment:
Find [tex]\( g \circ g \)[/tex]

Sagot :

GinnyAnsweridn
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]
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