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 12 x and g left parenthesis x right parenthesis equals StartFraction x Over 12 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.)

Answer 1

To find \( f(g(x)) \), you need to substitute \( g(x) \) into \( f(x) \).

Given:

• \( f(x) = 12x \)
• \( g(x) = \frac{x}{12} \)

Now, let's find \( f(g(x)) \):

1. Substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\frac{x}{12}\right) \]

2. Now, apply the function \( f \) to \( \frac{x}{12} \): \[ f\left(\frac{x}{12}\right) = 12 \left(\frac{x}{12}\right) \]

3. Simplify: \[ f\left(\frac{x}{12}\right) = \frac{12x}{12} = x \]

So, \[ f(g(x)) = x \]

Part 1 Answer:

\( f(g(x)) = x \)

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Part 2

Now let's find \( g(f(x)) \):

1. Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(12x) \]

2. Now, apply the function \( g \) to \( 12x \): \[ g(12x) = \frac{12x}{12} \]

3. Simplify: \[ g(12x) = x \]

So, \[ g(f(x)) = x \]

Part 2 Answer:

\( g(f(x)) = x \)

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Part 3

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), we can conclude that the functions \( f(x) \) and \( g(x) \) are indeed inverse functions of each other.

Part 3 Conclusion:

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