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 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.