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1.5. Basics of Functions
The notion of a function is one of the most useful concepts in mathematics. Functions are used throughout all areas of math and science. We have already seen a few examples of functions, namely univariate polynomials and rational functions. In this section, we discuss more general properties of functions.
A function f may be thought of as a machine which, given an input x, yields a unique output, y = f(x). Specifically, corresponding to every real number x lying in the domain of f, there is another real number, y=f(x).
There are several important definitions associated with functions. The domain of a function f is the subset of real numbers x for which f(x) is defined. The domain of most functions we will consider is the entire set of real numbers, though there are important exceptions. The range of f is the set of all real numbers y = f(x), where x varies throughout the domain of f. The real number x is called the argument of the function f(x), the number y = f(x) is called the image of x under f, and x is called a preimage of y under f. Preimages may or may not be unique, but images are always unique. A function is said to be one-to-one if every preimage of the function is unique.
Example 1: Consider the function f(x) = 2x. What is f(5)? What is the preimage of 20? What are the domain and range of f? Is f one-to-one?
Solution: To compute f(5), we simply substitute 5 for x in the definition of f(x). Thus we find f(5) = 2*5 = 10. Another way to say this is that 10 is the image of 5 under f and that 5 is the preimage of 10 under f. What about the domain and range of f? Since f(x) = 2x can be computed for all x, we see that the domain of f is the entire set of real numbers. Also, since every real number is twice another real number, we see that the range of f is also the entire set of real numbers. We see that f is one-to-one since the preimage of every number y under f is y/2. Thus the preimage of 20 is 10.
Example 2: As a slightly more complicated example, consider the function f(x) = x2. What is f(-3)? What are the preimages of 9? What are the domain and range of f? Is f one-to-one?
Solution: Here we see that f(-3) = (-3)2 = 9. Thus, 9 is the image of -3 under f and -3 is a preimage of 9 under f. Note that this preimage is not unique, however, since f(3) is also equal to 9, so 3 is also a preimage of 9. Thus, f is not one-to-one. The domain of f is the entire set of real numbers, since every real number has a square. But the range of f is just the set of nonnegative real numbers, since squares of real numbers are never negative.
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