(a)
For an exponential function in the form \(\displaystyle{y}={a}\cdot{b}^{{x}}\),
If \(\displaystyle{b}{>}{1}\), the function is increasing and is an exponential growth
function.
If \(\displaystyle{0}{<}{b}{<}{1}\)</span>, the function is decreasing and is an exponential decay
function.
Because \(\displaystyle{b}={\frac{{{1}}}{{{2}}}}{<}{1}\)</span>, then the function represents an exponential decay.
(b)
In an exponential function \(\displaystyle{y}={a}\cdot{b}^{{x}}\), a represents the initial value so 128
means that there were initially 128 teams in the tournament.
(c)
The exponential decay function is given by:
\(\displaystyle{A}{\left({t}\right)}={a}{\left({1}-{r}\right)}^{{t}}\)
where a is the initial amount, \(\displaystyle{\left({1}-{r}\right)}\) is the decay factor, and r is the rate
of decay.
Using the value of \(\displaystyle{b},{b}={1}-{r}\rightarrow{\frac{{{1}}}{{{2}}}}={1}-{\frac{{{1}}}{{{2}}}}\)
So, the rate of decay is:
\(\displaystyle{r}={\frac{{{1}}}{{{2}}}}={0.5}{\quad\text{or}\quad}{50}\%\)
This means that 50% of the teams are eliminated after each round.
(d)
Both the round numbers and the number of teams must be positive integer
values and 1 winner is determined after 7 rounds. Hence, a reasonable
domain is the integer values from 0 to 7. A reasonable range is from 1 to
128.