Question

Find f left parenthesis g left parenthesis x right parenthesis right parenthesis and g left parenthesis f left parenthesis x right parenthesis right parenthesis and determine whether the pair of functions f and g are inverses of each other. Complete parts a through c below.
f left parenthesis x right parenthesis equals 17 x and g left parenthesis x right parenthesis equals StartFraction x Over 17 EndFraction
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Part 1
a.  f left parenthesis g left parenthesis x right parenthesis right parenthesisequals
  
enter your response here  ​(Simplify your​ answer.)
Part 2
b.  g left parenthesis f left parenthesis x right parenthesis right parenthesisequals
  
enter your response here ​ (Simplify your​ answer.)
Part 3
c. Are the two functions f and g inverses of each​ other?
f and g are inverses of each other.
f and g are not inverses of each other.

Answer 1

Let's start by calculating \( f(g(x)) \) and \( g(f(x)) \) for the given functions \( f(x) = 17x \) and \( g(x) = \frac{x}{17} \).

### Part 1: Find \( f(g(x)) \)

To find \( f(g(x)) \):
1. Substitute \( g(x) \) into \( f(x) \):
\[
g(x) = \frac{x}{17}
\]
Then,
\[
f(g(x)) = f\left(\frac{x}{17}\right) = 17\left(\frac{x}{17}\right)
\]
2. Simplify:
\[
f\left(\frac{x}{17}\right) = 17 \cdot \frac{x}{17} = x
\]

Thus,
\[
\boxed{x}
\]

### Part 2: Find \( g(f(x)) \)

To find \( g(f(x)) \):
1. Substitute \( f(x) \) into \( g(x) \):
\[
f(x) = 17x
\]
Then,
\[
g(f(x)) = g(17x) = \frac{17x}{17}
\]
2. Simplify:
\[
g(17x) = 17x \cdot \frac{1}{17} = x
\]

Thus,
\[
\boxed{x}
\]

### Part 3: Determine if \( f \) and \( g \) are inverses

Since we have found:
- \( f(g(x)) = x \)
- \( g(f(x)) = x \)

This shows that:
\[
f(g(x)) = x \quad \text{and} \quad g(f(x)) = x
\]

Therefore, \( f \) and \( g \) are inverses of each other.

Thus, the answer is:
\[
\text{f and g are inverses of each other.}
\]

You can conclude with:
\[
\text{f and g are inverses of each other.}
\]