70
CHAPTER 1
=
__
___________
FUNCTIONS AND MODELS
function such that f(2)
3 + x + er, find g’(4)
I
9, what
x + .r + x, flndf(3) and f(f(2)).
f is a one-to-one
is f’(9)?
15. [f
16. Iff(x)
17. If g(x)
18. The graph of f is given.
(a) Why is f one-to-one?
(b) What are the domain and range of
(c) What is the value of f’(2)?
(d) Estimate the value of f(O).
—
32), where F
—459.67, expresses
19. The formula C = 4(F
the Celsius temperature C as a function of the Fahrenheit
temperature F. Find a formula for the inverse function and
interpret it. What is the domain of the inverse function’?
=
1(v)
=
Jl —V-/c-
1fl0
20. In the theory of relativity, the mass of a particle with speed
is
U
in
’
2
e
1 + 2 + 3x
26. v
24. y
=
=
22. fC)
=
I
—
x,
x
2x + 3
4x
—
2e
e’
+
4
where iO is the rest mass of the particle and c is the speed
of light in a vacuum. Find the inverse function off and
explain its meaning.
InC + 3)
=
21—26 Find a formula for the inverse of the function.
21. f(x)
=
23. f(x)
25. v
x>0
28. f(x)’2—e’
27—28 Find an explicit formula for f
1 and use it to graph frn’,
screen. To check your work,
f, and the line v = x on the same,
I
are reflections about the
see whether the graphs of .f and 1
line.
27. f(x)=x
+ 1,
4
29—30 Use the given graph off to sketch the graph of f’.
30
—
2
1.
0
x
31. Let 1(x) = x/l
(a) Findf’. How is it related to f?
(b) Identify the graph of land explain your answer to part (a).
How is the logarithmic function y = log,x defined?
What is the domain of this function’?
What is the range of this function?
Sketch the general shape of the graph of the function
y=log,,xifa> I.
.
3
32. Letg=/1 —x
(a) Find g. How is it related tog?
(b) Graph g. How do you explain your answer to part (a)?
33. (a)
(b)
(c)
(d)
34. (a) What is the natural logarithm?
(b) What is the common logarithm?
(c) Sketch the graphs of the natural logarithm function and
the natural exponential function with a common set of
axes.
(b) log3()
35—38 Find the exact value of each expression.
35. (a) Iog5 125
10
(b) tog
/1ö
36. (a) ln(l/e)
—
(b) ln(ln e’’)
2 20
log2 15 + log
37. (a) 1og26
(b) Iog3 100— log 18— 1og350
5
38. (a) 2
e
—
b)
—
2 In c
39—41 Express the given quantity as a single logarithm.
39. 1n5+51n3
40. ln(a + b) + ln(a
+
nx_lnsinx
4
+x2)
41. ln(1 l
42. Use Formula 10 to evaluate each logarithm correct to six
decimal places.
(b) 1og28.4
(a) logI2 10
=
v
=
In x,
v=logiox,
logi.sx,
=
v
=
v=lOr
x,
10
log
v=e,
y
x
50
log
43—44 Use Formula 10 to graph the given functions on a common
screen. How are these graphs related?
43. y
44.v=lnx,
and g(x)
=
lnx by graph-
x is drawn on a coordinate
2
45. Suppose that the graph of)’ = log
grid where the unit of measurement is an inch. How many
miles to the right of the origin do we have to move before the
height of the curve reaches 3 ft?
46. Compare the functions f(x)
48. (a)
V
ln(—x)
=
6
I)
I
=
3
49—52 Solve each equation for.v.
49. (a) ’
74
e
—
3
50. (a) ln(x
2
51. (a) 2’
=
=
52. (a) ln(ln x)
—
(b)y=ln\
—
(b) ln(3.v
(b) e
’
1
(h) lnx+
(b) c’’
(b) Ini>
53—54 Solve each inequality for x.
53. (a) e’< 10
3
’
2
(b) e
56.
f()
=
f
54. (a)
<lnx<9
2
=
55—56 Find (a) the domain off and (b)
55.1(i)
57. Graph the funciionf() = x/.v’+ x
2 +
why it is one-to-one. Then use a comput
to find an explicit expression forf’().
produce three possible expressions. ExpI
are irrelevant in this context.)
+
am em
(x, )
—
=
Cu es
(.Rt. g(t))
eter) I
Imagi
descri
the x
and i
gives i
SLIJ
58. (a) If gCr) = r
6 + .ï, .r
0, use a coml
to find an expression for g(x).
(h) Use the expression in part (a) to graf
and v = g 1() on the same screen.
.‘
FIGURE 1