The functions are at it again! There is another race to infinity. The functions $g(x) = 10x^2$ and $h(x) = 1.01^x$ are both in the race. A third mystery function with the values shown in the table is also in the race. The mystery function arrives late, so it does not get to the starting line until after the race has already begun.

1. Which function will be ahead when $x = 10$? Explain.

   Hint (a):

   Substitute $10$ for $x$ in each of the three functions. Which is the greatest?

2. How “fast” is each function going during the first two seconds of the race? That is, what is the average rate of change of each function for the interval $0 ≤ x ≤ 2$?

   Hint (b):

   Use the equations below:

   Avg. rate of change of  $g(x) = \frac{g(2)-g(0)}{2-0}$

   Avg. rate of change of  $h(x) = \frac{h(2)-h(0)}{2-0}$

   Avg. rate of change of  $m(x) =\frac{m(2)-m(0)}{2-0}$

3. Write a possible equation for the mystery function.

   Answer (c):

   $m(x) = 0.01(2)^x$

4. Which function will eventually win the race? When will it take the lead?

   Hint (d):

   Compare $g(x)$ and $m(x)$. Which grows faster, $x^2$ or $2^x$? Try substituting a very large number for $x$.

   Answer (d):

   The mystery function, $m(x)$, will win the race. It will take the lead between $x = 18$ and $x = 19$.