We have seen that different elements in the domain of a function may have the same image.If images are always different, then, as in the next definition, the function is called one-to-one.Tags: Dessay Fille Du RgimentEssay Unity Of MuslimNorthwestern Creative Writing CertificateEssays On Paradise LostDissertation Abstracts International PublishedPremium Writing PaperHoliday HomeworkBeef Cattle Business Plan
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).
When two variables r and A are related in this manner, it is customary to use the phrase A is a function of r.
To cite another example, if an automobile travels at a uniform rate of 50 miles per hour, then the distance d (miles) traveled in time t (hours) is given by d = 50t and hence the distance d is a function of time t.
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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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).