Understanding and reasoning with functions permeates mathematics education standards at all grade levels. Research shows students take multiple approaches to building functions. Deep content knowledge of functions may affect how pre-service teachers link student reasoning with mathematical ideas. To extend and connect the research literature on pre-service teachers' understanding of functions, this study elaborates on how pre-service teachers reason with different families of functions. This dissertation study investigates pre-service mathematics teachers' reasoning with linear, quadratic, and exponential functions in a variety of representations. This study was conceptualized within an algebraic and functional thinking framework. The research questions guiding this study are the following: (1) In what ways do pre-service teachers demonstrate their reasoning with functions? (2) How do various representations of function tasks influence pre-service teachers' reasoning with functions? The data analyzed in this study were from six semi-structured task-based clinical interviews conducted with three pre-service mathematics teachers. The results show pre-service mathematics teachers used multiple approaches to create rules for linear, quadratic, and exponential functions. The pre-service teachers demonstrated their reasoning with functions through five major strategies: covariation, correspondence, decomposition, composition, and graphical structure. The results also show that the representation of a function can influence pre-service teachers' approach. Visualization and composition played important roles in growth pattern tasks. Graphical structure could lead to function identification when prototypical features of the function were salient in the graph. Covariation, correspondence, and decomposition were often used with tables. Tables were also frequently created by pre-service teachers while engaging with tasks displaying functions in other representations. [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.]