Find $$ f(g(x)) $$ and $$ g(f(x)) $$ and determine whether the pair of functions $$ f $$ and $$ g $$ are inverses of each other $$ f(x)=4 x+2 $$ and $$ g(x)=\frac{x-2}{4} $$ a. $$ f(g(x))=x $$ (Simplify your answer) b. $$ g(f(x))=\square $$ (Simplify your answer)

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Let's determine $$ f(g(x)) $$ and $$ g(f(x)) $$ for the given functions and verify whether $$ f $$ and $$ g $$ are inverses of each other.

**Given Functions:**
$$
f(x) = 4x + 2 \
g(x) = \frac{x - 2}{4}
$$

### a. Compute $$ f(g(x)) $$

$$
f(g(x)) = f\left(\frac{x - 2}{4}\right)
$$
$$
= 4 \left( \frac{x - 2}{4} \right) + 2
$$
$$
= (x - 2) + 2
$$
$$
= x
$$

So,
$$
f(g(x)) = x
$$

### b. Compute $$ g(f(x)) $$

$$
g(f(x)) = g(4x + 2)
$$
$$
= \frac{(4x + 2) - 2}{4}
$$
$$
= \frac{4x}{4}
$$
$$
= x
$$

So,
$$
g(f(x)) = x
$$

### Conclusion: Are $$ f $$ and $$ g $$ Inverses?

Since both compositions satisfy:
$$
f(g(x)) = x \quad \text{and} \quad g(f(x)) = x
$$
this means that $$ f $$ and $$ g $$ are indeed **inverse functions** of each other.

**Answer Summary:**
- **a.** $$ f(g(x)) = x $$
- **b.** $$ g(f(x)) = x $$

Since both compositions equal $$ x $$, $$ f $$ and $$ g $$ are inverses of one another.
$$ f(g(x)) = x $$ and $$ g(f(x)) = x $$. Therefore, $$ f $$ and $$ g $$ are inverses of each other.

Find \( f(g(x)) \) and \( g(f(x)) \) and determine whether the pair of functions \( f \) and \( g \) are inverses of each other \( f(x)=4 x+2 \) and \( g(x)=\frac{x-2}{4} \) a. \( f(g(x))=x \) (Simplify your answer) b. \( g(f(x))=\square \) (Simplify your answer)

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