# Exponential functions on the real numbers

Evaluate and interpret exponential functions at non-integer inputs(N-RN.A.1, F-LE.A.2$^\star$).

In Exponential Functions 1 students considered exponential functions at integer inputs only. Now that they understand how to determine the value of $b^x$ for any rational number $x$, they can approximate $b^x$ to any degree of accuracy for any real number $x$. In this section they broaden their view of exponential functions to include the entire real number line as a possible domain. They evaluate functions and interpret their values at real inputs in terms of a context, in preparation for the more sophisticated work in the following sections.

## Tasks

WHAT: Students are presented with a situation where 20 mg of an allergy medication is administered, and the amount in the bloodstream decreases by 15% each hour. They are prompted to construct a function in the form $f(t) = ab^x$ to model the situation and then use it to find how much of the drug remains after different time intervals by evaluating $f(24)$, $f(1/2)$ and $f(1/60)$ (MP.4). There are opportunities to evaluate and interpret a rational exponent. Results from the model are approximations and the task does not specify to what place students should round; both of these offer opportunities for students to attend to precision (MP.6).

WHY: When students first start studying exponential functions they evaluate them at integer inputs. After they have learned about rational exponents they can evaluate them at other inputs. The purpose of the task is to help students become accustomed to evaluating exponential functions at non-integer inputs and interpreting the values.

WHAT: Students are told that the population of Pittsburgh was about 12,600 in 1820, and grew exponentially between 1820 and 1840 at a rate of 70% per decade. They are ask to construct a function in the form $f(t) = ab^t$ to model the population as a function of the number of decades after 1820. Then they evaluate $f(0.5)$ to find the population in 1825, $f(1.9)$ to find the population in 1839, and finally investigate the percent increase between 1826 and 1836. The observation that the population grew by 70% in this decade provides an opportunity to discuss the nature of exponential functionsâ€”that they change by a constant factor over any equal interval. <\p>

WHY: The purpose of this task is to give students experience working with simple exponential models in situations where they must evaluate and interpret them at non-integer inputs. Although not a proof, this task reinforces the understanding behind, that exponential functions grow by equal factors over equal intervals, even when those intervals do not have integer endpoints. Since students end up evaluating the expression $12600(1.7)^t$ for several values of $t$, this is an opportunity to use available tools (MP.5) to avoid typing the same expression repeatedly.

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