*After all, you wouldn’t want a surgeon to crack your ribs and perform a heart transplant without first identifying the source of your chest pains. Now that you understand the word problem’s purpose, determine the answer’s unit.*For example, will the answer be in miles, feet, ounces, pesos, dollars, the number of trees, or a number of televisions?The notion of correspondence is encountered frequently in everyday life.

Thus to 3 we assign 9, to - 5 we assign 25, and so on. All the examples of correspondences we have given are functions, as defined below.

Definition A function f from a set X to a set Y is a correspondence that assigns to each element x of X a unique element y of Y.

Beginning students are sometimes confused by the symbols f and f(x). Two functions f and g from X to Y are said to be equal, written for every x in X.

Example 1 Let f be the function with domain R such that f(x) = x, where a is a real number.

If a function is defined as in the preceding example, the symbol used for the variable is immaterial; that is, expressions such as: and so on, all define the same function.

This is true because if a is any number in the domain of f, then the same image a is obtained no matter which expression is employed. Solution As in Example 1, finding images under f is simply a matter of substituting the appropriate number for x in the expression for f(x).Since the square of any real number is nonnegative.T is contained in the set of all nonnegative real numbers.Moreover, every nonnegative real number c is an image underf, since .Hence the range of f is the set of all nonnegative real numbers.The element y is called the image of x under f and is denoted by f(x). The range of the function consists of all images of elements of X.Earlier, we introduced the notation f(x) for the element of Y which corresponds to x.Example 2 Let X denote the set of nonnegative real numbers and let f be the function from X to R defined by for every x in X. Thus: Many formulas which occur in mathematics and the sciences determine functions.As an illustration, the formula A = pi*r, and we may write A= f(r).The letter r, which represents an arbitrary number from the domain off, is often called an independent variable.The letter A, which represents a number from the range off, is called a dependent variable, since its value depends on the number assigned tor.

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