College Algebra 9th txtbk

SECTION 3–2 Graphing Functions 181
We used a solid dot at the point (2, 3) to indicate that y 4x 11 and y 3
agree at x 2. The solid dot at the point (1, 3) indicates that y 3 and
y 1 2 x 7 2 agree at x 1.
(C) The domain of a piecewise-defined function is the union of the intervals used in its
definition:
Domain of f: (, 2) ´ [2, 1] ´ (1, ) (, )
The graph of f shows that the range of f is (, 3]. The function f is increasing on
(, 2), constant on [2, 1], and decreasing on (1, ).
MATCHED PROBLEM 4
The function f is defined by
(A) Find f(4), f(1), f(3), and f(4).
f (x) 1 3 x 7 3 if x 1
2 if 1 6 x 6 3
5x 17 if x 3
(B) Graph f.
(C) Find the domain, range, and intervals where f is increasing, decreasing, or constant.
Notice that the graph of f in Example 4 contains no breaks. Informally, a graph (or portion
of a graph) is said to be continuous if it contains no breaks or gaps. (A formal presentation
of continuity can be found in calculus texts.)
Piecewise-defined functions occur naturally in many applications, especially ones
involving money. A very useful example is income tax.
EXAMPLE 5 Income Tax
Table 1 contains a recent tax rate chart for a single filer in the state of Oregon. If T(x) is
the tax on an income of $x, write a piecewise definition for T. Find the tax on each of the
following incomes: $2,000, $5,000, and $9,000.
Table 1 2009 Tax Rate Chart for Persons Filing Single, or Married Filing Separately
If the taxable income is:
The tax is:
Not over $3,050
5% of taxable income
Over $3,050 but not over $7,600 $153 plus 7% of the excess over $3,050
Over $7,600 $471 plus 9% of the excess over $7,600
Source: Oregon Department of Revenue
SOLUTION
Since taxes are computed differently on [0, 3,050], (3,050, 7,600] and (7,600, ), we must find
an expression for the tax on incomes in each of these intervals.
[0, 3,050]: Tax is 0.05x.
(3,050, 7,600]: Tax is $153 0.07(x 3,050) 0.07x – 61*
(7,600, ): Tax is $471 0.09(x 7,600) 0.09x 213
*In the Oregon tax rate chart, dollar amounts ending with 0.50 were rounded up to the next dollar. We will do
the same.