To determine whether two functions are inverse functions, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \).
Let's analyze each pair:
a.
- \( f(x) = x - 4 \)
- \( g(x) = x + 4 \)
Calculating \( f(g(x)) \): \[ f(g(x)) = f(x + 4) = (x + 4) - 4 = x \]
Calculating \( g(f(x)) \): \[ g(f(x)) = g(x - 4) = (x - 4) + 4 = x \]
Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), \( f(x) \) and \( g(x) \) are inverse functions.
b.
- \( f(x) = x - 4 \)
- \( g(x) = \frac{x - 4}{4} \)
Calculating \( f(g(x)) \): \[ f(g(x)) = f\left(\frac{x - 4}{4}\right) = \frac{x - 4}{4} - 4 = \frac{x - 4 - 16}{4} = \frac{x - 20}{4} \] This is not equal to \( x \).
Calculating \( g(f(x)) \): \[ g(f(x)) = g(x - 4) = \frac{(x - 4) - 4}{4} = \frac{x - 8}{4} \] This is also not equal to \( x \).
Thus, \( f(x) \) and \( g(x) \) are not inverse functions.
c.
- \( f(x) = x - 4 \)
- \( g(x) = 4x - 1 \)
Calculating \( f(g(x)) \): \[ f(g(x)) = f(4x - 1) = (4x - 1) - 4 = 4x - 5 \] This is not equal to \( x \).
Calculating \( g(f(x)) \): \[ g(f(x)) = g(x - 4) = 4(x - 4) - 1 = 4x - 16 - 1 = 4x - 17 \] This is also not equal to \( x \).
Thus, \( f(x) \) and \( g(x) \) are not inverse functions.
Conclusion:
The only pair of functions that are inverse functions is: a. \( f(x) = x - 4 \) and \( g(x) = x + 4 \).