# Problem Solving Algebraic Expressions 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.

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.

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The notion of correspondence is encountered frequently in everyday life.

For example, to each book in a library there corresponds the number of pages in the book.

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.

The examples of correspondences we have given involve two sets X and Y.

In our first example, X denotes the set of books in a library and Y the set of positive integers.

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.