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Using the two functions listed below, insert numbers in place of the letters a, b, c, and d so that f(x) and g(x) are inverses.

a. f(x)=(x+a)/b
b. g(x) = cx-d

1 Answer

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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.

User Orochi
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