The purpose of this study was to determine (1) strategies students use when solving composition problems and the difficulties they encounter; (2) conceptions and/or misconceptions students have with respect to composition of functions; and (3) the effect of using dynamic visualization during instruction on students' understanding of composition of functions. This study employed a mixed-methods design. The qualitative phase of the study included one-on-one interviews with seventeen of the pre-calculus students. These interviews lasted between fifteen minutes and one hour and sought to uncover student's understanding about function composition and function inverse by investigating their approaches taken when solving the pre-, post- and second post-test items. From these interviews, common themes emerged and assertions related to student understanding were developed. The quantitative phase of the study included pre-, post- and second post-tests designed to determine the effect of the dynamically enhanced instruction on student's performance. There were six test items, including four function composition items and two function inverse items. The centerpiece of this instruction included a flash applet designed to illustrate function composition in three different representations. The study began with 183 pre-calculus students divided into two groups--control and treatment. Using the test results, statistical analyses were conducted and conclusions were made regarding differences in student performance by group. The major findings included: (1) Students mistook the small composition symbol for multiplication. (2) When composing functions numerically, students read the table in the wrong direction. (3) Students with an understanding of function input and output performed better on composition items than those who did not. (4) Students with an understanding of the association between the y-axis and points on a graph performed better than those who did not. (a) During interviews, treatment students exhibited a better understanding of this y-axis awareness than control students and performed better on the graphical composition items. (5) Students performed worse on function inverse items than composition items and understanding function composition did not seem to be a support for understanding function inverse. (6) Generally speaking, students did not have a conceptual understanding of function inverse; for example, the undoing aspect of inverse functions was not understood. (7) Treatment students performed better on a composition task involving the graphical representation using unlabeled graphs. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]