*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.*

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.

You can review your answers and change them by checking the desired letter.

Once you have finished, press "finish" and you get a table with your answers and the right answers to compare with.

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).

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