Nonroutine function tasks are more challenging than most typical high school mathematics tasks. Nonroutine tasks encourage students to expand their thinking about functions and their approaches to problem solving. As a result, they gain greater appreciation for the power of multiple representations and a richer understanding of functions. This article presents four tasks wherein students are asked to investigate algebraic and graphical representations of several types of functions and are thus provided opportunities to develop, use, and compare a variety of solution strategies. Some of these solution methods take advantage of less-used but nonetheless valuable features of graphing calculators to generate representations for learners to analyze. Appropriate use of these tasks challenges learners' intuitions while broadening and deepening their knowledge of functions through the linking of multiple representations. Although these tasks have been around for a while, most students find them to be novel, engaging, and thought provoking. These tasks support NCTM's Algebra, Representation, Connections, and Problem Solving Standards for grades 9-12 as well as the Learning and Technology Principles. In addition, the tasks address topics assessed by the 2005 National Assessment of Educational Progress, such as identifying and graphing exponential functions, solving routine and nonroutine problems involving different representations of functions, and performing operations on algebraic expressions using appropriate tools. (Contains 5 figures.)