Find $$ f(g(x)) $$ and $$ g(f(x)) $$ and determine whether the pair of functions $$ f $$ and $$ g $$ are inverses of each other. $$ f(x)=3 x $$ and $$ g(x)=\frac{x}{3} $$ a. $$ f(g(x))=x $$ b. $$ g(f(x))=\square $$

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UpStudy AI · Accepted Answer

To determine whether the functions $$ f(x) = 3x $$ and $$ g(x) = \frac{x}{3} $$ are inverses of each other, we'll compute the compositions $$ f(g(x)) $$ and $$ g(f(x)) $$.

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

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

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

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

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

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

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

Two functions $$ f $$ and $$ g $$ are **inverse functions** of each other if and only if:

1. $$ f(g(x)) = x $$ for all $$ x $$ in the domain of $$ g $$.
2. $$ g(f(x)) = x $$ for all $$ x $$ in the domain of $$ f $$.

From our calculations:

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

Both compositions simplify to $$ x $$, satisfying the conditions for inverse functions. Therefore, **$$ f $$ and $$ g $$ are indeed inverses of each other**.

### Verification

To further verify, we can check if applying one function cancels out the other:

- **Applying $$ f $$ then $$ g $$:**
  $$
  g(f(x)) = g(3x) = \frac{3x}{3} = x
  $$
  
- **Applying $$ g $$ then $$ f $$:**
  $$
  f(g(x)) = f\left(\frac{x}{3}\right) = 3 \times \frac{x}{3} = x
  $$
  
Both operations return the original input $$ x $$, confirming that $$ f $$ and $$ g $$ are inverses.

### Final Answers

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

**Conclusion:** $$ f $$ and $$ g $$ are inverse functions of each other.
$$ f(g(x)) = x $$ 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. \( f(x)=3 x \) and \( g(x)=\frac{x}{3} \) a. \( f(g(x))=x \) b. \( g(f(x))=\square \)

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