Final answer:
To make the functions f(x) and g(x) inverses, we need b = c = 1 and a = d. This ensures that when we compose the functions, we obtain the identity function, confirming their inverse relationship.
Step-by-step explanation:
To find the values of a, b, c, and d such that the functions f(x) = (x + a) / b and g(x) = cx - d are inverses of each other, we need to apply the concept that two functions are inverses if and only if composing them results in the identity function, which is f(g(x)) = x and g(f(x)) = x.
Substituting g(x) into f(x):
f(g(x)) = f(cx - d) = ((cx - d) + a) / b = x
Multiplying both sides by b to eliminate the denominator:
b * x = (cx - d) + a
Furthermore, because f(g(x)) must simplify to x, we require that:
b = 1 since b * x should equal x
c = 1 because cx must ultimately be x
a = d so that -d + a = 0 and does not alter the value of x.
Therefore, a proper choice for the constants would be a = d and b = c = 1, making the functions inverses of each other.