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 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.