Functions have an important place in the Swedish national curriculum. This study explores Swedish high school students’ conceptions and reasoning when solving mathematical problems involving functions. Two students were asked to solve three problems and reason aloud while describing how they approached their solutions. These problems dealt with function expressions, function representations, and the definition of the function concept. The study adopted a discourse analysis, where recorded discussions and explanations were examined to ascertain students’ strategies, abilities, challenges, conceptual understanding, and misconceptions. The main insights from the study are based on exploring how misconceptions and conceptual understanding underlie students’ challenges and abilities. Understanding representations seemed to increase general abilities, and the presence of correct conceptual definitions did not imply a good understanding of the concept of functions. The students’ algebraic knowledge did not hinder their function understanding, but rather was helpful when moving between representations and handling function expressions.