Question

If a bacteria population starts with 80 bacteria and doubles every four hours, then the number of bacteria after t hours is

n = f(t) = 80 · 2t⁄4.
Find the inverse of this function.

Answer 1

Final answer:

The inverse function is
`\(f^(-1)(n) = 4 \cdot \log_2\left((n)/(80)\right)\)`.

Step-by-step explanation:

The given function is
`\(n = f(t) = 80 \cdot 2^(t/4)\)`, representing the number of bacteria after
`\(t\)` hours. To find the inverse, interchange
`\(n\)` and
`\(t\)` and solve for
`\(t\)`.

1. Swap
`\(n\)` and
`\(t)`:

`\[ t = 80 \cdot 2^(n/4) \]`

2. Solve for
`\(n)`:

`\[ (t)/(80) = 2^(n/4) \]`

3. Take the logarithm of both sides to isolate the exponent:

`\[ \log_2\left((t)/(80)\right) = (n)/(4) \]`

4. Multiply both sides by 4 to solve for
`\(n)`:

`\[ n = 4 \cdot \log_2\left((t)/(80)\right) \]`

So, the inverse function is
`\(f^(-1)(n) = 4 \cdot \log_2\left((n)/(80)\right)\)`.