before noon. These times are reasonable, considering the power consumption schedules of most individuals and
businesses.
24. Runner A won the race, reaching the finish line at 100 meters in about 15 seconds, followed by runner B with a time of about
19 seconds, and then by runner C who finished in around 23 seconds. B initially led the race, followed by C, and then A.
C then passed B to lead for a while. Then A passed first B, and then passed C to take the lead and finish first. Finally,
B passed C to finish in second place. All three runners completed the race.
25. Of course, this graph depends strongly on the
geographical location!
26. The summer solstice (the longest day of the year) is
around June 21, and the winter solstice (the shortest day)
is around December 22. (Exchange the dates for the
southern hemisphere.)
27. As the price increases, the amount sold decreases.
28. The value of the car decreases fairly rapidly initially, then
somewhat less rapidly.
29.
c ° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.
SECTION 1.1 FOURWAYS TO REPRESENTA FUNCTION
¤
13
30. (a) (b)
(c) (d)
31. (a) (b) 9:00 AM corresponds to = 9. When = 9, the
temperature is about 74 ◦ F.
32. (a) (b) The blood alcohol concentration rises rapidly, then slowly
decreases to near zero.
33. () = 3 2 − + 2.
(2) = 3(2) 2 − 2 + 2 = 12 − 2 + 2 = 12.
(−2) = 3(−2) 2 − (−2) + 2 = 12 + 2 + 2 = 16.
() = 3 2 − + 2.
(−) = 3(−) 2 − (−) + 2 = 3 2 + + 2.
( + 1) = 3( + 1) 2 − ( + 1) + 2 = 3( 2 + 2 + 1) − − 1 + 2 = 3 2 + 6 + 3 − + 1 = 3 2 + 5 + 4.
2() = 2 · () = 2(3 2 − + 2) = 6 2 − 2 + 4.
(2) = 3(2) 2 − (2) + 2 = 3(4 2 ) − 2 + 2 = 12 2 − 2 + 2.
[continued]
c ° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.
14
¤
CHAPTER 1 FUNCTIONSAND MODELS
( 2 ) = 3( 2 ) 2 − ( 2 ) + 2 = 3( 4 ) − 2 + 2 = 3 4 − 2 + 2.
[()] 2 =
3 2
− + 2 2 =
3 2
− + 2 3 2 − + 2
= 9 4 − 3 3 + 6 2 − 3 3 + 2 − 2 + 6 2 − 2 + 4 = 9 4 − 6 3 + 13 2 − 4 + 4
( + ) = 3( + ) 2 − ( + ) + 2 = 3( 2 + 2 + 2 ) − − + 2 = 3 2 + 6 + 3 2 − − + 2.
34. () =
√ + 1 .
(0) =
0
√ 0 + 1 = 0.
(3) =
3
√ 3 + 1 =
3
2 .
5() = 5 ·
√ + 1 =
5
√ + 1 .
1
2 (4) =
1
2
· (4) =
1
2
·
4
√ 4 + 1 =
2
√ 4 + 1 .
( 2 ) =
2
√ 2
+ 1 ; [()]
2
=
√ + 1
2
=
2
+ 1 .
( + ) =
( + )
( + ) + 1 =
+
√ + + 1 .
( − ) =
( − )
( − ) + 1 =
−
√ − + 1 .
35. () = 4 + 3 − 2 , so (3 + ) = 4 + 3(3 + ) − (3 + ) 2 = 4 + 9 + 3 − (9 + 6 + 2 ) = 4 − 3 − 2 ,
and
(3 + ) − (3)
=
(4 − 3 − 2 ) − 4
=
(−3 − )
= −3 − .
36. () = 3 , so ( + ) = ( + ) 3 = 3 + 3 2 + 3 2 + 3 ,
and
( + ) − ()
=
( 3 + 3 2 + 3 2 + 3 ) − 3
=
(3 2 + 3 + 2 )
= 3 2 + 3 + 2 .
37. () =
1
, so
() − ()
−
=
1
−
1
−
=
−
−
=
−
( − )
=
−1( − )
( − )
= −
1
.
38. () =
√ + 2, so () − (1)
− 1