Answer:
Initial Value is [tex]f(t) = \frac{2}{3}[/tex], This function follows exponential decay
Step-by-step explanation:
The initial value of any time based function is the value of f(t) at t = 0. For calculating the initial value of this function all we have to do is insert 0 as the value of t.
[tex]f(0) = \frac{2}{3} (1)\\ f(0) = \frac{2}{3}[/tex]
Hence the initial value is [tex]\frac{2}{3}[/tex]
To find out whether this function is exponential growth or decay we need to ascertain whether the base value of the power t is greater than or lesser than 1
In this case the base value of [tex](\frac{1}{3} )^t[/tex] is [tex]\frac{1}{3}[/tex] which is lesser than 1, hence this function is exponential decay since with each increase in power the total value will decrease,i.e.
[tex]=(\frac{1}{3} )^0 = 1\\\ = (\frac{1}{3})^1 = 0.333\\\ =(\frac{1}{3})^2 = 0.1111\\ =(\frac{1}{3})^3 = 0.037\\ .\\ .\\ .\\[/tex]
This can also be proven from the graph below
