## A.2 Functions Very simply, a ***function** f* of one variable is a rule or law that associates with a value *x* a unique value *f* (*x*). For example, the function *f* that associates the square of a real number with a given real number *x* is  A function determines a set of ordered pairs. For example, the function *f* (*x*) = *x*2 determines all the ordered pairs (*x*, *x*2). A ***graph*** of a function is the set of all ordered pairs determined by the function. The graph of the function *f* (*x*) = *x*2 appears in [Figure A.1](#ap-afig01). [](figap-a-1_0.jpg) Figure A.1: The graph of the function *f* (*x*) = *x*2. The ordered pair (2, 4) is illustrated. The function,  is defined only if *x* ≠ 0. The ***domain*** of a function is the set of values for which the function is defined. For example, the domain of *f* (*x*) = 1/*x* is all real numbers other than 0, whereas the domain of *f* (*x*) = *x*2 is all real numbers. Notice that the function  can assume only nonnegative values. By "nonnegative values" we mean values greater than or equal to 0, whereas by "positive values" we mean values strictly greater than 0. The ***range*** of a function is the set of values that the function can assume. The range of *f* (*x*) = *x*2 is the nonnegative reals, the range of *f* (*x*) = 1/*x* is all real numbers other than 0, and the range of *f* (*x*) = (1/*x*)2 is all positive reals. We say that a function is *from* its domain and *to* its range. For example, the function *f* (*x*) = *x*2 is from the reals to the nonnegative reals.