The function f(x) = 81(b)^x, where b is a positive constant, is given. For every increase in the value of x by 1, the value of f(x) increases by c%, where 0 < c < 100. Which expression gives the value of c in terms of b?
(A) 1 + \frac{b}{100}
(B) b + 100
(C) 100(b + 1)
(D) 100(b - 1)
Answer (2)
To find the expression for the percentage increase c in terms of b , let's analyze the function f ( x ) = 81 b x . The question states that for every increase in x by 1, f ( x ) increases by c % .
Here’s how we can find c :
Set up the relationship for f ( x + 1 ) : The function at x + 1 is f ( x + 1 ) = 81 b x + 1 = 81 b x ⋅ b .
Calculate the growth factor: Dividing f ( x + 1 ) by f ( x ) gives: f ( x ) f ( x + 1 ) = 81 b x 81 b x ⋅ b = b . This means that f ( x + 1 ) is b times f ( x ) .
Relate b to c : If f ( x ) increases by c % , then: 1 + 100 c = b . Solving for c , we get: c = 100 ( b − 1 ) .
This corresponds to option D. Therefore, the expression that gives the value of c in terms of b is:
Option D) 100 ( b − 1 ) .
The expression for the percentage increase c in the function f ( x ) = 81 b x in terms of b is given by c = 100 ( b − 1 ) . Thus, the correct choice is (D).
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