Answer:
[tex]f^{-1}=3ln_2(\frac{t}{100})[/tex]
f^-1 represents the number of hours the bacteria developed
Step-by-step explanation:
[tex]f(t) = 100 \cdot 2^\frac{t}{3}[/tex]
To find inverse of a function , replace f(t) with y
[tex]y= 100 \cdot 2^\frac{t}{3}[/tex]
replace t with y and y with x
[tex]x= 100 \cdot 2^\frac{y}{3}[/tex]
solve for y
divide both sides by 100
[tex]\frac{x}{100} =2^\frac{y}{3}[/tex]
take ln on both sides
[tex]ln \frac{x}{100} =\frac{y}{3}ln 2[/tex]
divide both sides by ln
[tex]ln_2(\frac{x}{100}) =\frac{y}{3}[/tex]
multiply by 3 on both sides
[tex]y=3ln_2(\frac{x}{100})[/tex]
Replace x with t and y with f^-1
[tex]f^{-1}=3ln_2(\frac{t}{100})[/tex]
f^-1 represents the number of hours the bacteria developed
