Functions Worksheet With Answer Key Page 21
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4689
2.3 PIECEWISE-DEFINED FUNCTIONS
Problems
17. Figure 2.22 shows an entire graph. An open circle repre-
1 for x < 0
20. Let g(x) =
sents a point that is not included.
3
x
for x ≥ 0
(a) Is y a function of x? Explain.
(b) Is x a function of y? Explain.
(a) Find g( 2), g(2), and g(0).
(c) If you identified a function in part (a) or part (b),
(b) Find the domain and range of g(x).
what is the range of that function?
y
3x
for
1 ≤ x ≤ 1
4
21. Let f (x) =
3
x + 4 for 1 < x ≤ 5
2
1
(a) Find f (0) and f (3).
x
(b) Find the domain and range of f (x).
1
2
3
4
Figure 2.22
1/x for x <
1
√
x
2
= x. We will investigate
18. Many people believe that
22. Let f (x) =
2
x
for
1 ≤ x ≤ 1
√
this claim graphically and numerically.
x
for x > 1.
√
(a) Graph the two functions x and
x
2
in the window
5 ≤ x ≤ 5,
5 ≤ y ≤ 5. Based on what you see,
(a) Evaluate f ( 2) and f (2).
√
x
2
= x? What function does
do you believe that
(b) What is the range of f ?
√
x
2
the graph of
remind you of?
(b) Complete Table 2.11. Based on this table, do you
√
23. Use bracket notation to write a formula for the piecewise
believe that
x
2
= x? What function does the ta-
√
2
function that is defined by y = x
x
2
for negative values of
ble for
remind you of? Is this the same function
x and by y = x
1 for x values that are greater than or
you found in part (a)?
equal to zero.
Table 2.11
x
5
4
3
2
1
0
1
2
3
4
5
24. A floor-refinishing company charges $1.83 per square
√
foot to strip and refinish a tile floor for up to 1000 square
x
2
feet. There is an additional charge of $350 for toxic waste
√
disposal for any job that includes more than 150 square
(c) Explain how you know that
x
2
is the same as the
feet of tile.
function |x|.
√
x
2
|x| in the window
5 ≤
(d) Graph the function
(a) Express the cost, y, of refinishing a floor as a func-
x ≤ 5, 5 ≤ y ≤ 5. Explain what you see.
tion of the number of square feet, x, to be refinished.
19. (a) Graph u(x) = |x|/x in the window
5 ≤ x ≤ 5,
(b) Graph the function. Give the domain and range.
5 ≤ y ≤ 5. Explain what you see.
(b) Complete Table 2.12. Does this table agree with
what you found in part (a)?
25. The charge for a taxi ride in New York City is $2.50 upon
entry and $0.40 for each 1/5 of a mile traveled (rounded
Table 2.12
up to the nearest 1/5 mile), when the taxicab is travel-
ing at 6 mph or more. In addition, a New York State Tax
x
5
4
3
2
1
0
1
2
3
4
5
10
Surcharge of $0.50 is added to the fare.
|x|/x
(a) Make a table showing the cost of a trip as a function
(c) Identify the domain and range of u(x).
of its length. Your table should start at zero and go
(d) Comment on the claim that u(x) can be written as
up to two miles in 1/5-mile intervals.
1 if x < 0,
(b) What is the cost for a 1.2-mile trip?
u(x) =
(c) How far can you go for $5.80?
0 if x = 0,
(d) Graph the cost function in part (a).
1 if x > 0.
10
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