Textbook Chapter 1

Section 1.2 Functions and Graphs 15
EXAMPLE 3 Identifying Domain and Range of a Function
Use a grapher to identify the domain and range, and then draw a graph of the function.
(a) y 4 x 2
(b) y x 2/3
SOLUTION
(a) Figure 1.14a shows a graph of the function for 4.7 x 4.7 and
3.1 y 3.1, that is, the viewing window [4.7, 4.7] by [3.1, 3.1], with
x-scale y-scale 1. The graph appears to be the upper half of a circle. The domain
appears to be [2, 2]. This observation is correct because we must have
4 x 2 0, or equivalently, 2 x 2. The range appears to be [0, 2], which can
also be verified algebraically.
y = x 2/3
Graphing y = x 2/3 —–Possible
Grapher Failure
On some graphing calculators you need
to enter this function as y (x 2 ) 1/3 or
y (x 1/3 ) 2 to obtain a correct graph.
Try graphing this function on your grapher.
y = 4 – x 2 [–4.7, 4.7] by [–2, 4]
[–4.7, 4.7] by [–3.1, 3.1]
(a)
(b)
Figure 1.14 The graph of (a) y 4 x 2 and (b) y x 2/3 . (Example 3)
y
(b) Figure 1.14b shows a graph of the function in the viewing window
[4.7, 4.7] by [2, 4], with x-scale y-scale 1. The domain appears to be
(, ), which we can verify by observing that x 2/3 ( 3
x) 2 . Also the range is
[0, ) by the same observation. Now try Exercise 15.
y x 2 (x, y)
(–x, y)
O
x
Even Functions and Odd Functions—Symmetry
The graphs of even and odd functions have important symmetry properties.
(a)
y
O
y x 3 (x, y)
x
DEFINITIONS Even Function, Odd Function
A function y ƒ(x) is an
even function of x if f x f x,
odd function of x if f x f x,
for every x in the function’s domain.
(–x, –y)
(b)
Figure 1.15 (a) The graph of y x 2
(an even function) is symmetric about the
y-axis. (b) The graph of y x 3 (an odd
function) is symmetric about the origin.
The names even and odd come from powers of x. If y is an even power of x, as in y x 2 or
y x 4 , it is an even function of x (because (x) 2 x 2 and (x) 4 x 4 ). If y is an odd power
of x, as in y x or y x 3 , it is an odd function of x (because (x) 1 x and (x) 3 x 3 ).
The graph of an even function is symmetric about the y-axis. Since f (x) f (x), a point
(x, y) lies on the graph if and only if the point (x, y) lies on the graph (Figure 1.15a).
The graph of an odd function is symmetric about the origin. Since f (x) = f (x), a
point (x, y) lies on the graph if and only if the point (x, y) lies on the graph (Figure 1.15b).