Chapter 2 Functions

Functions provide a precise way to describe how one quantity depends on another. In water engineering, a function may connect time to tank volume, flow to pressure loss, concentration to sensor output, or pipe diameter to cross-sectional area. A function is more than a formula. Its domain states which inputs are permitted, its range states which outputs occur, and its graph shows how the output changes. This chapter follows the progression of the MATH 128 notes from function notation and graphs through rates of change, transformations, combinations, composition, one-to-one and onto behaviour, and inverse functions. Throughout the chapter, units and physical restrictions are treated as part of the model.

Learning outcomes. By the end of this chapter, you should be able to:

• evaluate functions and interpret their inputs and outputs;
• determine domains and ranges from formulas and graphs;
• analyze piecewise functions, net change, and average rate of change;
• solve equations and inequalities graphically and describe graph behaviour;
• graph functions using transformations;
• combine and compose functions while tracking domains; and
• determine whether a function is one-to-one or onto and find an inverse when one exists.