Questions: For each pair of functions f and g below, find f(g(x)) and g(f(x)). Then, determine whether f and g are inverses of each other. Simplify your answers as much as possible. (Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain.) (a) f(x) = 1/(2x), x ≠ 0 (b) f(x) = x + 6 g(x) = 1/(2x), x ≠ 0 f(g(x)) = g(f(x)) = f and g are inverses of each other f and g are inverses of each other f and g are not inverses of each other f and g are not inverses of each other

Solution Steps

To solve the problem, we need to find the compositions \( f(g(x)) \) and \( g(f(x)) \) for each pair of functions \( f \) and \( g \). Then, we check if these compositions result in the identity function \( x \), which would indicate that \( f \) and \( g \) are inverses of each other.

1. For the first pair of functions \( f(x) = \frac{1}{2x} \) and \( g(x) = \frac{1}{2x} \):

   • Compute \( f(g(x)) \) by substituting \( g(x) \) into \( f(x) \).
   • Compute \( g(f(x)) \) by substituting \( f(x) \) into \( g(x) \).
   • Check if both compositions simplify to \( x \).

2. For the second pair of functions \( f(x) = x + 6 \) and \( g(x) = \frac{1}{2x} \):

   • Compute \( f(g(x)) \) by substituting \( g(x) \) into \( f(x) \).
   • Compute \( g(f(x)) \) by substituting \( f(x) \) into \( g(x) \).
   • Check if both compositions simplify to \( x \).

Given the functions:

• \( f(x) = \frac{1}{2x} \)
• \( g(x) = \frac{1}{2x} \)

We compute: \[ f(g(x)) = f\left(\frac{1}{2x}\right) = \frac{1}{2 \cdot \frac{1}{2x}} = x \] \[ g(f(x)) = g\left(\frac{1}{2x}\right) = \frac{1}{2 \cdot \frac{1}{2x}} = x \]

Both compositions simplify to \( x \), indicating that \( f \) and \( g \) are inverses of each other.

Given the functions:

• \( f(x) = x + 6 \)
• \( g(x) = \frac{1}{2x} \)

We compute: \[ f(g(x)) = f\left(\frac{1}{2x}\right) = \frac{1}{2x} + 6 = 6 + \frac{1}{2x} \] \[ g(f(x)) = g(x + 6) = \frac{1}{2(x + 6)} \]

The compositions do not simplify to \( x \), indicating that \( f \) and \( g \) are not inverses of each other.

Final Answer