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Example 4: Functions Described by Explicitly Giving Its Domain

Let f be the function given by the rule

f(t) = 01_real_and_hyperreal_numbers-33.gif

Thus f(x) is the positive square root of 1 - x2. The domain of f is the closed interval [-1, 1]. The range of f is [0, 1].

For instance,

f(-2) is undefined,         f(-1) = 0,         f(0) = 1,

f(½) = 01_real_and_hyperreal_numbers-34.gif.          f(1) = 0,         f(2) is undefined.

The graph of f is given by the equation

y = 01_real_and_hyperreal_numbers-35.gif

The equation can also be written in the form

x2 + y2 = 1,   y ≥ 0.

The graph is just the upper half of the unit semicircle, shown in Figure 1.2.6.


Figure 1.2.6

Sometimes a function is described by explicitly giving its domain in addition to a rule.

Last Update: 2006-11-14