# Solve Algebraic Problems

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.

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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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