Textbook Chapter 1

44 Chapter 1 Prerequisites for Calculus
Section 1.5 Exercises
35. x ln 3 5
2
0.96 or 0.96 36. x log 2 5
221
2.26 or 2.26
In Exercises 1–6, determine whether the function is one-to-one.
1. y
2.
y
No
Yes
3. y
4.
Yes
No
5. y
6.
Yes
No
y = –3x 3
0
y = 1 x
y 2 x
In Exercises 7–12, determine whether the function has an inverse
function.
3
7. y 1 Yes 8. y x x 2
2 5x No 9. y x 3 4x 6 No
10. y x 3 x Yes 11. y ln x 2 No 12. y 2 3x Yes
In Exercises 13–24, find f 1 and verify that
x
x
x
y = int x
( f f 1 )(x) ( f 1 f )(x) x.
y
y
y x + 1
x
1
y =
x 2 + 1
x
x
In Exercises 33–36, solve the equation algebraically. Support your
solution graphically.
ln
2
33. (1.045) t t
2 34. e 0.05t ln 3
ln 1.045
3 t 21.97
0 .05
35. e x e x 3 15.75 36. 2 x 2 x 5
In Exercises 37 and 38, solve for y.
37. ln y 2t 4 y e 2t4 38. ln (y 1) ln 2 x ln x
y 2xe x 1
In Exercises 39–42, draw the graph and determine the domain and
range of the function.
Domain: (, 3); Range: all reals Domain: (2, ); Range: all reals
39. y 2 ln (3 x) 4 40. y 3 log (x 2) 1
Domain: (1, ); Range: all reals Domain: (4, ); Range: all reals
41. y log 2 (x 1) 42. y log 3 (x 4)
In Exercises 43 and 44, find a formula for f 1 and verify that
( f f 1 )(x) ( f 1 ƒ)(x) x.
f 1 (x) log 1.1
x
50 x
100
50
43. f (x) 1 2x
44. f (x)
1 1.1 x
45. Self-inverse Prove that the function f is its own inverse.
(a) f (x) 1 x 2 , x 0 (b) f (x) 1x
46. Radioactive Decay The half-life of a certain radioactive substance
is 12 hours. There are 8 grams present initially.
(a) Express the amount of substance remaining as a function of
time t. Amount 8 1 2 t/12
(b) When will there be 1 gram remaining? After 36 hours
47. Doubling Your Money Determine how much time is required
for a $500 investment to double in value if interest is earned at
the rate of 4.75% compounded annually. About 14.936 years.
(If the interest is only paid annually, it will take 15 years.)
48. Population Growth The population of Glenbrook is 375,000
and is increasing at the rate of 2.25% per year. Predict when the
population will be 1 million. After about 44.081 years
In Exercises 49 and 50, let x 0 represent 1990, x 1 represent
1991, and so forth.
49. Natural Gas Production
(a) Find a natural logarithm regression equation for the data in
Table 1.16 and superimpose its graph on a scatter plot of the data.
13. f (x) 2x 3 f 1 (x) x 3
Table 1.16 Canada’s Natural Gas
14. f (x) 5 4x f 1 (x) 5 x
2
4
Production
15. ƒ(x) x 3 1 16. f (x) x 2 1, x 0
17. f (x) x 2 f
, x 0 1 (x) x 1/2
Year Cubic Feet (trillions)
18. f (x) x 23 , x 0
or x
f 1 (x) x 3/2
1997 5.76
19. f (x) (x 2) 2 , x 2 f 1 (x) 2 (x) 1/2 or 2 x
1998 5.98
20. f (x) x 2 2x 1, x 1 f 1 (x) x 1/2 1 or x 1 21. f 1 1
(x) x
1/2 or
1x 1999 6.26
1
21. f (x) x 2, x 0 22. f (x) x
13
2000 6.47
f 1 1 1
(x) x
1/3 or
3
x
2001 6.60
23. f (x) 2 x 1
24. f (x) x
3
Source: Statistical Abstract of the United States,
f
x 3
x 2
1 (x) 2 x 3
f 1 (x) 1 3x
2004–2005.
x 2
x 1
In Exercises 25–32, use parametric graphing to graph f, f 1 , and y x.
(b) Estimate the number of cubic feet of natural gas produced by
Canada in 2002. Compare with the actual amount of 6.63 trillion
25. f (x) e x 26. f (x) 3 x 27. f (x) 2 x
cubic feet in 2002. 6.79 trillion; the estimate exceeds the actual
28. f (x) 3 x 29. f (x) ln x 30. f (x) log x
amount by 0.16 trillion cubic feet
(c) Predict when Canadian natural gas production will reach 7
31. f (x) sin 1 x 32. f (x) tan 1 x
trillion cubic feet. sometime during 2003
15. f 1 (x) (x 1) 1/3 or 3 x 1 16. f 1 (x) (x 1) 1/2 or x 1
43. f 1 (x) log 2
x
100 x