Example 4: Functions Described by Explicitly Giving Its Domain

Let f be the function given by the rule

f(t) =

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

The graph of f is given by the equation

y =

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.