Watch the video and then solve the problem given below. Click here to watch the video. Find $$ f(g(x)) $$ and $$ g(f(x)) $$ and determine whether the pair of functions $$ f $$ and $$ g $$ are inverses of each other. $$ f\left(x+1\right. $$ and $$ g(x)=\frac{x-1}{3} $$ a. $$ f(g(x))=x $$ (Simplify your answer.) b. $$ g(f(x))=x $$ (Simplify your answer.) c. Are $$ f $$ and $$ g $$ inverses of each other? Yes No

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To solve the problem, we need to find $$ f(g(x)) $$ and $$ g(f(x)) $$ using the given functions $$ f(x) = x + 1 $$ and $$ g(x) = \frac{x - 1}{3} $$.

### Step 1: Find $$ f(g(x)) $$

1. Substitute $$ g(x) $$ into $$ f(x) $$:
   $$
   f(g(x)) = f\left(\frac{x - 1}{3}\right)
   $$
2. Now, apply the function $$ f $$:
   $$
   f\left(\frac{x - 1}{3}\right) = \left(\frac{x - 1}{3}\right) + 1
   $$
3. Simplify the expression:
   $$
   f(g(x)) = \frac{x - 1}{3} + 1 = \frac{x - 1 + 3}{3} = \frac{x + 2}{3}
   $$

### Step 2: Find $$ g(f(x)) $$

1. Substitute $$ f(x) $$ into $$ g(x) $$:
   $$
   g(f(x)) = g(x + 1)
   $$
2. Now, apply the function $$ g $$:
   $$
   g(x + 1) = \frac{(x + 1) - 1}{3} = \frac{x}{3}
   $$

### Step 3: Determine if $$ f $$ and $$ g $$ are inverses

To check if $$ f $$ and $$ g $$ are inverses, we need to see if:
- $$ f(g(x)) = x $$
- $$ g(f(x)) = x $$

From our calculations:
- $$ f(g(x)) = \frac{x + 2}{3} $$ (not equal to $$ x $$)
- $$ g(f(x)) = \frac{x}{3} $$ (not equal to $$ x $$)

Since neither $$ f(g(x)) $$ nor $$ g(f(x)) $$ equals $$ x $$, we conclude that $$ f $$ and $$ g $$ are not inverses of each other.

### Final Answers

a. $$ f(g(x)) = \frac{x + 2}{3} $$

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

c. Are $$ f $$ and $$ g $$ inverses of each other? 
**No**
$$ f(g(x)) = \frac{x + 2}{3} $$, $$ g(f(x)) = \frac{x}{3} $$, and $$ f $$ and $$ g $$ are not inverses of each other.

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