Which is the initial value that shrinks an exponential growth function by $50 \%$?
A. $\frac{1}{5}$
B. $\frac{1}{4}$
C. $\frac{1}{3}$
D. $\frac{1}{2}$
Answer (1)
The problem asks for a value that shrinks an exponential growth function by 50%.
Let the exponential growth function be f ( x ) = a × b x .
We want to find c such that c × f ( x ) = 0.5 × f ( x ) .
Solving for c , we find c = 2 1 .
Explanation
Understanding the Problem We are looking for a value that, when multiplied by an exponential growth function, reduces the function's value by 50%. In other words, we want to find a value c such that multiplying an exponential growth function by c is the same as multiplying it by 0.5 .
Setting up the Equation Let the exponential growth function be represented as f ( x ) = a × b x , where a is the initial value and 1"> b > 1 is the growth factor. We want to find a value c such that c × f ( x ) = 0.5 × f ( x ) .
Isolating the Unknown This means c × a × b x = 0.5 × a × b x . Since a and b x are non-zero, we can divide both sides of the equation by a × b x to isolate c .
Finding the Value Dividing both sides by a × b x , we get c = 0.5 . Since 0.5 = f r a c 1 2 , the value that shrinks the exponential growth function by 50% is f r a c 1 2 .
Final Answer Therefore, the initial value that shrinks the exponential growth function by 50% is f r a c 1 2 .
Examples
Imagine you're baking a cake, and the recipe calls for a certain amount of sugar. If you want to make the cake only half as sweet, you would multiply the amount of sugar by 2 1 . Similarly, in exponential growth, multiplying the initial value by 2 1 reduces the entire function by 50% at every point. This concept is useful in various applications, such as adjusting medication dosages, controlling population growth in models, or scaling down financial investments.
