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The curved arrows indicate that the elements f(x), f(w), f(z), and f(a) of Y correspond to the elements x, y, z and a of X.Let us repeat the important fact that to each x in X there is assigned precisely one image f(x) in Y; however, different elements of X such as w and z in Figure 1.18 may have the same image in Y. However, f(x) is an element of Y, namely the element which f assigns to x.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.
Think of the calculator as merely a tool that makes the journey easier.
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).
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.