Functions

Functions model situations where one quantity determines another. For example, the return on $10,000 invested at an annualized percentage rate of 4.25% is a function of the length of time the money is invested. Because nature and society are full of dependencies between quantities, functions are important tools in the construction of mathematical models.

In school mathematics, functions usually have numerical inputs and outputs and are often defined by an algebraic expression. For example, the time in hours it takes for a plane to fly 1000 miles is a function of the plane's average ground speed in miles per hour, //v//; the rule //T//(//v//) = 1000///v// expresses this relationship algebraically and defines a function whose name is //T//.

The set of possible inputs to a function is called its domain. We often infer the domain to be all inputs for which the expression defining a function has a value, or for which the function makes sense in a given context. The graph of a function is a useful way of visualizing the relationship the function models, and manipulating the expression for a function can throw light on the function's properties.

Two important families of functions characterized by laws of growth are linear functions, which grow at a constant rate, and exponential functions, which grow at a constant percent rate. Linear functions with an initial value of zero describe proportional relationships.

//Connections to Expressions, Equations, Modeling and Coordinates.// Determining an output value for a particular input involves evaluating an expression; finding inputs that yield a given output involves solving an equation. The graph of a function //f// is the same as the solution set of the equation //y// = //f//(//x//). Questions about when two functions have the same value lead to equations, whose solutions can be visualized from the intersection of the graphs. Since functions describe relationships between quantities, they are frequently used in modeling. Sometimes functions are defined by a recursive process, which can be modeled effectively using a spreadsheet or other technology.

Students understand that: Students can and do:
 * Core Concepts**
 * 1) A function is a rule, often defined by an expression, that assigns a unique output for every input.
 * 2) The graph of a function //f// is a set of ordered pairs (//x//, //f//(//x//)) in the coordinate plane.
 * 3) Functions model situations where one quantity determines another.
 * 4) Common functions occur in families where each member describes a similar type of dependence.
 * Core Skills**
 * 1) Recognize proportional relationships and solve problems involving rates and ratios.
 * 2) Describe the qualitative behavior of common types of functions using graphs and tables.
 * 3) Analyze functions using symbolic manipulation.
 * 4) Use the families of linear and exponential functions to solve problems.
 * 5) Find and interpret rates of change.