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. $$ f(x)=3 x+1 $$ and $$ g(x)=\frac{x-1}{3} $$ a. $$ f(g(x))=\square $$ (Simplify your answer.)

Question

UpStudy AI · Accepted Answer

**Step 1: Compute $$ f(g(x)) $$**

Given:
$$
f(x) = 3x + 1 \quad \text{and} \quad g(x) = \frac{x-1}{3}
$$

Substitute $$ g(x) $$ into $$ f $$:
$$
f(g(x)) = f\left(\frac{x-1}{3}\right)
$$

Using the definition of $$ f $$:
$$
f\left(\frac{x-1}{3}\right) = 3\left(\frac{x-1}{3}\right) + 1
$$

Simplify the expression:
$$
3\left(\frac{x-1}{3}\right) = x - 1
$$
$$
x - 1 + 1 = x
$$

Thus, the simplified result is:
$$
f(g(x)) = x
$$

**Step 2: Compute $$ g(f(x)) $$**

Substitute $$ f(x) $$ into $$ g $$:
$$
g(f(x)) = g(3x+1)
$$

Using the definition of $$ g $$:
$$
g(3x+1) = \frac{(3x+1)-1}{3}
$$

Simplify the expression:
$$
(3x+1)-1 = 3x
$$
$$
\frac{3x}{3} = x
$$

Thus, the simplified result is:
$$
g(f(x)) = x
$$

**Conclusion:**

Since both compositions yield:
$$
f(g(x)) = x \quad \text{and} \quad g(f(x)) = x
$$
the functions $$ f $$ and $$ g $$ are indeed inverses of each other.

**Final Answers:**

a. $$ f(g(x)) = x $$
$$ f(g(x)) = x $$

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