Functions are one of the most important topics in secondary school mathematics, especially for students who wish to take higher-level mathematics courses beginning with calculus. The prerequisites for Advanced Placement Calculus state that a thorough understanding of functions is needed for those who wish to succeed in the course and pass the AP Calculus Exam. However, research has shown that students' struggles with calculus concepts could be traced to inadequate prerequisite knowledge of functions. Since thousands of students take and pass the AP Calculus Exam every year, it is important to investigate which aspects of functions AP Calculus students generally do and do not understand, and to determine how well their understandings of functions relate to their performance on the exam. In order to explore this, students from AP Calculus classes in three different schools were tested on their understandings of functions at the end of the course, after they had already taken the AP Calculus Exam. Additionally, some of the participants were also interviewed to further ascertain their understandings of functions. Finally, the AP Calculus Exam scores for all participants were collected and compared to their function understandings. It was found that 1) most participants' understandings of functions were less than sufficient for an AP Calculus course, and 2) there was a high positive correlation between function understandings and scores on the AP Calculus Exam. These results suggest that more work must be done in developing students' understandings of functions in the secondary mathematics curriculum, and greater measures should be taken to ensure that students entering AP Calculus have a sufficient understanding of the prerequisite knowledge. [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://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]