CHAPTER 5
Functions and their Graphs
In Problems 5.1-5.19, find the domain and range, and draw the graph, of the function determined by the given formula.
5.1
h(x) = 4-x
2 .
5.2
5.3
5.4
5.5
The domain consists of all real numbers, since 4 — x
2
is defined for all x. The range consists of all real
numbers < 4: solving the equation y = 4 — x
2
for x, we obtain x = ±\/4 — y, which is defined when and
only when y < 4. The graph (Fig. 5-1) is a parabola with vertex at (0, 4) and the y-axis as its axis of symmetry.
Fig. 5-1
G(x) = -2Vx.
The domain consists of all nonnegative real numbers. The range consists of all real numbers s 0. The graph
(Fig. 5-2) is the lower half of the parabola 4x = y
2
.
The domain is the closed interval [-2,2], since V4-x
2 is defined when and only when *
2 s4. The
graph (Fig. 5-3) is the upper half of the circle x
2 + y
2 - 4 with center at the origin and radius 2. The range is
the closed interval [0,2].
Fig. 5-3
Fig. 5-4
omain consists of all x such that x SL 2 or x ^ —2, since we must have x2 SL 4. The graph (Fig.I The domain consists of all x such that x SL 2 or x ^ —2, since we must have x2 SL 4. The graph (Fig.
5-4) is the part of the hyperbola x
2 -y
2 = 4 on or above the x-axis. The range consists of all nonnegative real
numbers.
V(x) = \x-l\.
The domain is the set of all real numbers. The range is the set of all nonnegative real numbers. The graph
(Fig. 5-5) is the graph of y = \x\ shifted one unit to the right.
23
Fig. 5-2
Fig.5.5