The Domain and the Range of a Function

In the preceding section, you learned that relations and functions are sets of ordered pairs representing a correspondence of elements of two sets X and Y. You start with an element from the first set and pair it off with an element from the second set according to a prescribed relation or rule. The starting set X is called the domain of the relation. The set of all the second elements in each ordered pair is called the range. The range may or may not be the whole set Y.

Example:A. Let X = {1, 2, 3, 4} and Y = {1, 2, 3, 4. 5, 6}. Define a function from X to Y by the equation y=x+2. Why is this a function? The function consists of the following ordered pairs: (1,3),(2,4),(3,5),(4,6). The domain is X, and the range is the set {3, 4, 5, 6}⊆Y. How was this range obtained?

B. If a function involves a pairing of numbers and the sets X and Y are not specified, the domain is taken to be the largest subject of the set of real numbers for which y can be computed and is real. Thus if a function is defined by the equation , then since only nonnegative values of x can have real square roots, then the domain is the set of all nonnegative real numbers. What is the range?

C. Consider the function defined by the equation . The value of y will not be real only if the denominator is zero. Thus means that x cannot be -1, and the domain consists of all real numbers except -1. Can you determine the range? Hint: Try solving x in terms of y and check what value(s) for y is(are) impossible. Can you give a reason for which such values cannot be attained y?

Determine the domain and the range of each of the functions described below.1. 2.3.4.5.

Exercises: