Question

Geraldine is asked to explain the limits on the range of an exponential equation using the function f(x) = 2x. She makes these two statements:

1. As x increases infinitely, the y-values are continually doubled for each single increase in x.
2. As x decreases infinitely, the y-values are continually halved for each single decrease in x.

She concludes that there are no limits within the set of real numbers on the range of this exponential function. Which best explains the accuracy of Geraldine’s statements and her conclusion?

Answer 1

Geraldine’s qualitative descriptions match the behavior of the exponential function f(x) = 2^x, but her conclusion is slightly wrong.

Correct facts for f(x) = 2^x:
- If x increases, f(x) doubles for each +1 increment (2^{x+1} = 2·2^x) and lim_{x→∞} 2^x = ∞ (no upper bound).
- If x decreases, f(x) is halved for each −1 decrement (2^{x-1} = 2^x/2) and lim_{x→−∞} 2^x = 0 (values get arbitrarily small but stay positive).

Conclusion:
- There is no upper limit (y → ∞), but there is a lower bound: 0 is an infimum that the function approaches but never attains. Thus the range is (0, ∞), not all real numbers.

(If the function had been f(x) = 2x (linear), her doubling/halving statements would be incorrect and the range would be all real numbers.)