Glven the functions, $$ f(x)=3 x-2 $$ and $$ g(x)=\frac{x+2}{3} $$, complete parts 1 and 2 . 1. Find $$ f(g(x)) $$ and $$ g(f(x)) $$. Include your work in your final answer. 2. Use complete sentences to explain the relationship that exists between the composition of the functions, $$ f(g(x)) $$ and $$ g(f(x)) $$.

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Certainly! Let's work through each part step by step.

### 1. Finding $$ f(g(x)) $$ and $$ g(f(x)) $$

**Given Functions:**
- $$ f(x) = 3x - 2 $$
- $$ g(x) = \frac{x + 2}{3} $$

**a. Finding $$ f(g(x)) $$:**

To find $$ f(g(x)) $$, we substitute $$ g(x) $$ into $$ f(x) $$.

$$
f(g(x)) = f\left(\frac{x + 2}{3}\right)
$$

Now, replace $$ x $$ in $$ f(x) $$ with $$ \frac{x + 2}{3} $$:

$$
f\left(\frac{x + 2}{3}\right) = 3\left(\frac{x + 2}{3}\right) - 2
$$

Simplify the expression:

$$
3 \times \frac{x + 2}{3} = x + 2
$$

So,

$$
f(g(x)) = (x + 2) - 2 = x
$$

**b. Finding $$ g(f(x)) $$:**

To find $$ g(f(x)) $$, we substitute $$ f(x) $$ into $$ g(x) $$.

$$
g(f(x)) = g(3x - 2)
$$

Now, replace $$ x $$ in $$ g(x) $$ with $$ 3x - 2 $$:

$$
g(3x - 2) = \frac{(3x - 2) + 2}{3}
$$

Simplify the expression:

$$
(3x - 2) + 2 = 3x
$$

So,

$$
g(f(x)) = \frac{3x}{3} = x
$$

**Summary:**

$$
f(g(x)) = x
$$
$$
g(f(x)) = x
$$

### 2. Explanation of the Relationship Between $$ f(g(x)) $$ and $$ g(f(x)) $$

Both compositions $$ f(g(x)) $$ and $$ g(f(x)) $$ result in the original input variable $$ x $$. This indicates that the functions $$ f(x) $$ and $$ g(x) $$ are **inverse functions** of each other. In other words, applying one function after the other returns the input value, effectively "undoing" each other's operations.

Specifically:
- When you apply $$ g(x) $$ first and then $$ f(x) $$, you get back to $$ x $$.
- Similarly, applying $$ f(x) $$ first and then $$ g(x) $$ also brings you back to $$ x $$.

This mutual inversion confirms that $$ f $$ and $$ g $$ are inverses, meaning they reverse each other's effects when composed.
$$ f(g(x)) = x $$ and $$ g(f(x)) = x $$. This means that $$ f $$ and $$ g $$ are inverse functions, as applying one after the other returns the original input.

Glven the functions, and , complete parts 1 and 2 .

Find and . Include your work in your final answer.

Use complete sentences to explain the relationship that exists between the composition of the functions,
and .

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Glven the functions, and , complete parts 1 and 2 . Find and . Include your work in your final answer. Use complete sentences to explain the relationship that exists between the composition of the functions, and .