Question

A national trivia tournament starts with 262,144 teams. With each round, there are half as many teams as before. The number of teams after any number of rounds, x, can be modeled with the following function.f(x)=262144(1/2)^xWhich statement compares the mathematical range and reasonable range of the function?1. Both the mathematical and reasonable ranges are limited to real numbers greater than 0 and less than or equal to 262,144.2. Both the mathematical and reasonable ranges are limited to whole numbers greater than 0 and less than or equal to 262,144.3. The mathematical range is all real numbers greater than 0. The reasonable range is all real numbers greater than 0 and less than or equal to 262,144.4. The mathematical range is all real numbers greater than 0. The reasonable range is all whole numbers greater than 0 and less than or equal to 262,144

Answer 1

The given function is

`f(x)=262,144((1)/(2))^x`

This is a decreasing exponential function, due to the base is less than 1. When the base is less than one, the function is decreasing and 1/2 is less than 1.

The range of this exponential function is "all numbers greater than zero" because a power can't result in a null result (nor negative), that is, we can't get zero (or negative number) from a power, no matter what base it has, this applies for every exponential function.

However, the reasonable range refers to those values that make sense to the problem, because not all of the values from the function are valid as a reasonable solution.

Therefore, the reasonable range

`(0,262144\rbrack`

Notice that the limit is 262,144, since that's the maximum number of teams possible.

The right choice is the "The mathematical range is all real numbers greater than 0. The reasonable range is all whole numbers greater than 0 and less than or equal to 262,144"