

Lesson 4
Functions
SMR Domain 1.3
Although the SMRs are organized into discrete content domains (e.g., algebra or calculus), prospective teachers should know that these domains cannot be rigidly separated. For example, the importance of the exponential function (SMR 1.3c) stems primarily from the fact that it is the unique solution of the differential equation f'(x) = f(x) with the initial condition f(0) = 1 (SMR 5.3f). It should be noted that because of this differential equation, the exponential function `e^x` shows up in the growth and decay problems of algebra textbooks.
The fundamental difference between polynomial functions and both exponential and logarithmic functions should be emphasized (SMR 1.3b, c). The overriding concern with a polynomial is to locate its roots and the roots of its derivative (to get the xintercepts as well as the "peaks" and "valleys" of its graph).
Please read carefully the following summary description of this topic. Don't worry if you don't yet understand it completely, or, if you have already studied this material, check your understanding with the practice problems at the end and move on to the next topic.