When students are working with hand held technology, such as graphics calculators, we usually only see the outcomes of their activities in the form of a contribution to a written solution of a mathematical problem. It is more difficult to capture their process of thinking or actions as they use the technology to solve a problem. In this paper we report on a case study that followed the progress of twelve first year university students as they solved twelve mathematical problems associated with the investigation of functions and their graphs. We used software, which we refer to as the key recorder software. This software runs in the background of the graphics calculator, discreetly capturing the students' keystrokes as they make use of the calculator. The aim of the research study described in this paper was to provide insights into the problem solving strategies of these students. Through a detailed analysis of their graphics calculator keystrokes and associated written solutions, we have proposed a classification of different strategies that identifies the effectiveness of the solution strategies and the efficiency of the use of technology. This classification has important implications for the teaching of functions and their graphs with such technology. (Contains 7 tables and 13 figures.)