Find $$ f(g(x)) $$ and $$ g(f(x)) $$ and determine whether the pair of functions $$ f $$ and $$ g $$ are inverses of each other. $$ f(x)=3 x-7 $$ and $$ g(x)=\frac{x+3}{7} $$ a. $$ f(g(x))=\square $$ (Simplify your answer. Use integers or fractions for any numbers in the expression.) b. $$ g(f(x))=\square $$ (Simplify your answer. Use integers or fractions for any numbers in the expression.) c. $$ 0 \quad f $$ and $$ g $$ are not inverses of each other. f and $$ g $$ are inverses of each other.

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UpStudy AI · Accepted Answer

Evaluate the expression by following steps:
- step0: Evaluate:
  $$\frac{\left(x+3\right)}{7}$$
- step1: Substitute:
  $$\frac{\left(3x-7+3\right)}{7}$$
- step2: Add the numbers:
  $$\frac{\left(3x-4\right)}{7}$$
- step3: Remove the parentheses:
  $$\frac{3x-4}{7}$$
Substitute $$ x=(x+3)/7 $$ into the expression $$ 3x-7 $$.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$3x-7$$
- step1: Substitute:
  $$3\times \frac{\left(x+3\right)}{7}-7$$
- step2: Remove the parentheses:
  $$3\times \frac{x+3}{7}-7$$
- step3: Multiply the terms:
  $$\frac{3\left(x+3\right)}{7}-7$$
- step4: Reduce fractions to a common denominator:
  $$\frac{3\left(x+3\right)}{7}-\frac{7\times 7}{7}$$
- step5: Transform the expression:
  $$\frac{3\left(x+3\right)-7\times 7}{7}$$
- step6: Multiply the terms:
  $$\frac{3x+9-7\times 7}{7}$$
- step7: Multiply the numbers:
  $$\frac{3x+9-49}{7}$$
- step8: Subtract the numbers:
  $$\frac{3x-40}{7}$$
Let's solve the problem step by step.

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

We start with the function $$ g(x) = \frac{x+3}{7} $$ and substitute it into $$ f(x) = 3x - 7 $$:

$$
f(g(x)) = f\left(\frac{x+3}{7}\right) = 3\left(\frac{x+3}{7}\right) - 7
$$

Now, simplifying this expression:

$$
f(g(x)) = \frac{3(x+3)}{7} - 7 = \frac{3x + 9}{7} - 7
$$

To combine the terms, we convert $$ 7 $$ into a fraction with a denominator of $$ 7 $$:

$$
f(g(x)) = \frac{3x + 9}{7} - \frac{49}{7} = \frac{3x + 9 - 49}{7} = \frac{3x - 40}{7}
$$

Thus, the answer is:

$$
f(g(x)) = \frac{3x - 40}{7}
$$

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

Next, we substitute $$ f(x) = 3x - 7 $$ into $$ g(x) $$:

$$
g(f(x)) = g(3x - 7) = \frac{(3x - 7) + 3}{7}
$$

Now, simplifying this expression:

$$
g(f(x)) = \frac{3x - 7 + 3}{7} = \frac{3x - 4}{7}
$$

Thus, the answer is:

$$
g(f(x)) = \frac{3x - 4}{7}
$$

### c. Determining if $$ f $$ and $$ g $$ are inverses

For $$ f $$ and $$ g $$ to be inverses of each other, we need to check if:

1. $$ f(g(x)) = x $$
2. $$ g(f(x)) = x $$

From our calculations:

- $$ f(g(x)) = \frac{3x - 40}{7} $$
- $$ g(f(x)) = \frac{3x - 4}{7} $$

Neither of these expressions equals $$ x $$. Therefore, we conclude that:

$$
f \text{ and } g \text{ are not inverses of each other.}
$$

### Final Answers

a. $$ f(g(x)) = \frac{3x - 40}{7} $$

b. $$ g(f(x)) = \frac{3x - 4}{7} $$

c. $$ 0 \quad f \text{ and } g \text{ are not inverses of each other.} $$
a. $$ f(g(x)) = \frac{3x - 40}{7} $$ b. $$ g(f(x)) = \frac{3x - 4}{7} $$ c. $$ f $$ and $$ g $$ are not inverses of each other.

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