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 7 x and g left parenthesis x right parenthesis equals StartFraction x Over 7 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

Given the functions

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

we will compute \( f(g(x)) \) and \( g(f(x)) \) and assess whether they are inverses of each other.

### Part 1
**a. Compute \( f(g(x)) \)**:

Substituting \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left(\frac{x}{7}\right) = 7\left(\frac{x}{7}\right)
\]

Simplifying:
\[
f(g(x)) = x
\]

### Part 2
**b. Compute \( g(f(x)) \)**:

Substituting \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(7x) = \frac{7x}{7}
\]

Simplifying:
\[
g(f(x)) = x
\]

### Part 3
**c. Are the two functions \( f \) and \( g \) inverses of each other?**

Two functions \( f \) and \( g \) are inverses of each other if:
\[
f(g(x)) = x \quad \text{and} \quad g(f(x)) = x
\]

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

Both conditions are satisfied.

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

So the answers are:
- Part 1: \( f(g(x)) = x \)
- Part 2: \( g(f(x)) = x \)
- Part 3: **f and g are inverses of each other.**