SECTION 1.

4 Exponential Functions 53

1.4 EXERCISES
x x
1–4 Use the Law of Exponents to rewrite and simplify the 9. y − 3 x, y − 10 x, y − ( 13 ) , y − ( 101 )
expression.
10. y − 0.9 x, y − 0.6 x, y − 0.3 x, y − 0.1x
4 23 1
1. (a) 28 (b) -
2 s
3
x4
11–16 Make a rough sketch of the graph of the function. Do not
2. (a) 8 4y3 (b) xs3x 2 d3
use a calculator. Just use the graphs given in Figures 3 and 13
and, if necessary, the transformations of Section 1.3.
s6y 3d 4
3. (a) b8 s2bd 4 (b)
2y 5 11.. y − 4 x 2 1
11 12. y − s0.5d x 21

x 2n ? x 3n21 sa sb
s 13. y − 22 2x 14. y − e | x |
4. (a) (b) - -
x n12 s
3
ab 15. y − 1 2 12 e2x 16. y − 2s1 2 e x d

5. (a) Write an equation that defines the exponential function 17. Starting with the graph of y − e x, write the equation of the
with base b . 0. graph that results from
(b) What is the domain of this function? (a) shifting 2 units downward.
(c) If b ± 1, what is the range of this function? (b) shifting 2 units to the right.
(d) Sketch the general shape of the graph of the exponential (c) reflecting about the x-axis.
function for each of the following cases. (d) reflecting about the y-axis.
(i) b . 1 (e) reflecting about the x-axis and then about the y-axis.
(ii) b − 1
(iii) 0 , b , 1 18. Starting with the graph of y − e x, find the equation of the
graph that results from
6. (a) How is the number e defined? (a) reflecting about the line y − 4.
(b) What is an approximate value for e? (b) reflecting about the line x − 2.
(c) What is the natural exponential function?
19–20 Find the domain of each function.
; 7–10 Graph the given functions on a common screen. How are
1 2 ex 11x
2

these graphs related? 19. (a) f sxd − (b) f sxd −

1 2 e12x
2
e cos x
7. y − 2 x, y − e x, y − 5 x, y − 20 x
20. (a) tstd − s10 t 2 100 (b) tstd − sinse t 2 1d
2x 2x
8. y − e ,x
y−e , y−8, x
y−8
54 CHAPTER 1 Functions and Models

21–22 Find the exponential function f sxd − Cb x whose graph (c) Graph the model from part (b) together with the scatter
is given. plot in part (a). Use the TRACE feature to determine
how long it takes for the bacteria count to double.
21. 22.
y y
(3, 24) (_1, 3)

© Sebastian Kaulitzki / Shutterstock.com

4
”1, 3 ’
(1, 6)

0 x 0 x

23. If
23 I f sxd − 5 x, show that G. lamblia
f sx 1 hd 2 f sxd
h
− 5x S D
5h 2 1
h
30. A bacteria culture starts with 500 bacteria and doubles in
size every half hour.
(a) How many bacteria are there after 3 hours?
24. Suppose you are offered a job that lasts one month. Which (b) How many bacteria are there after t hours?
of the following methods of payment do you prefer? (c) How many bacteria are there after 40 minutes?
I. One million dollars at the end of the month. ; (d) Graph the population function and estimate the time
II. One cent on the first day of the month, two cents on the for the population to reach 100,000.
second day, four cents on the third day, and, in general,
2 n21 cents on the nth day. 31. The half-life of bismuth-210, 210 Bi, is 5 days.
(a) If a sample has a mass of 200 mg, find the amount
25. Suppose the graphs of f sxd − x 2 and tsxd − 2 x are drawn remaining after 15 days.
on a coordinate grid where the unit of measurement is (b) Find the amount remaining after t days.
1 inch. Show that, at a distance 2 ft to the right of the origin, (c) Estimate the amount remaining after 3 weeks.
the height of the graph of f is 48 ft but the height of the ; (d) Use a graph to estimate the time required for the mass
graph of t is about 265 mi. to be reduced to 1 mg.
; 26. Compare the functions f sxd − x and tsxd − 5 by graph- 32. An isotope of sodium, 24 Na, has a half-life of 15 hours. A
5 x

ing both functions in several viewing rectangles. Find all sample of this isotope has mass 2 g.
points of intersection of the graphs correct to one decimal (a) Find the amount remaining after 60 hours.
place. Which function grows more rapidly when x is large? (b) Find the amount remaining after t hours.
(c) Estimate the amount remaining after 4 days.
; 27. Compare the functions f sxd − x and tsxd − e by
10 x

graphing both f and t in several viewing rectangles. When ; (d) Use a graph to estimate the time required for the mass
to be reduced to 0.01 g.
does the graph of t finally surpass the graph of f ?
33. Use the graph of V in Figure 11 to estimate the half-life
; 28. Use a graph to estimate the values of x such that
of the viral load of patient 303 during the first month of
e x . 1,000,000,000.
treatment.
; 29. A researcher is trying to determine the doubling time for
34. After alcohol is fully absorbed into the body, it is metabo-
a population of the bacterium Giardia lamblia. He starts
lized with a half-life of about 1.5 hours. Suppose you have
a culture in a nutrient solution and estimates the bacteria
had three alcoholic drinks and an hour later, at midnight,
count every four hours. His data are shown in the table.
your blood alcohol concentration (BAC) is 0.6 mgymL.
(a) Find an exponential decay model for your BAC t hours
Time (hours) 0 4 8 12 16 20 24 after midnight.
Bacteria count ; (b) Graph your BAC and use the graph to determine when
sCFUymLd
I I I I I I I I
37 47 63 78 105 130 173 you can drive home if the legal limit is 0.08 mgymL.
Source: Adapted from P. Wilkinson et al., “Pharmacokinetics of Ethanol after
(a) Make a scatter plot of the data. Oral Administration in the Fasting State,” Journal of Pharmacokinetics and
Biopharmaceutics 5 (1977): 207–24.
(b) Use a graphing calculator to find an exponential curve
f std − a ? b t that models the bacteria population ; 35. Use a graphing calculator with exponential regression
t hours later. capability to model the population of the world with the