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.

Question

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