Question

If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after t hours is n = f(t) = 100 · 2t/3. Find the inverse of this function.f −1(n) =?

Answer 1

The inverse of the function n = f(t) = 100 · 2^{t/3} is found by taking the logarithm of both sides and solving for t. The inverse function is f^{-1}(n) = 3 · log(n/100) / log(2).

Step-by-step explanation:

To find the inverse of the function n = f(t) = 100 · 2^{t/3}, you need to solve for t in terms of n. The steps are as follows:

• First, divide both sides of the equation by 100 to isolate the exponential term:
  n/100 = 2^{t/3}.
• Next, take the logarithm of both sides to remove the exponent, using the fact that log(a^b) = b · log(a). You will have log(n/100) = (t/3) · log(2).
• Finally, solve for t by multiplying both sides of the equation by 3 and then dividing by log(2): t = 3 · log(n/100) / log(2).

Therefore, the inverse function is f^{-1}(n) = 3 · log(n/100) / log(2).