1. The graph f(x) = x2+x and g(x) = 2x+2 is below, left. The solution to f(x) = g(x) is x = -1, and x = 2. The solution to the inequality f(x) > g(x) is (-¥,-1)U(2,¥).
1. The graph f(x) = x2+x and g(x) = 2x+2 is below,
left.
The solution to f(x) = g(x) is x = -1, and x = 2. The solution
to the inequality f(x) > g(x) is (-¥,-1)U(2,¥).
2. The graph of g(x) = (x+1)4, for -4 < x £-2 is
above, right. The solution to g(x) = 25 is x = -4
Ö 25 - 1 = - Ö 5 -
1,
g-1(16) = -3, and g-1(x) = - 4 Ö x
- 1 for 1 £ x < 81.
3. The graph of
f(x) = (x -5)/(x
2
-2x -3). is
below.
The x- intercept is 5 and the y- intercept is 5/3. The
vertical asymptotes are x= -1 and
x= 3. The horizontal asymptote is y=
0. The domain of
log[(x -5)/(x
2
-2x -3)] is
(-1,3) U(5,¥).
4. The graph of f(x) = ex+1-4 is below, left. The asymptote is y = -4.
The y-intercept exact is e-4 which is -1.28 accurate to 2 decimal places:
the x-intercept exact is ln(4)-1 which is 0.39 accurate to 2 decimal places.
f -1(x) = ln(x+4)-1.
5. The graph f(x) = ln(x+3)-1 is above, right. The asymptote is x = -3. The y-intercept exact is ln(3)-1 which is 0.10 accurate to 2 decimal places: the x-intercept exact is e-3 which is - 0.28 accurate to 2 decimal places. f -1(x) = ex+1-3.
6. The domain of g(x) = 2ln(x)+3ln(x+4) is (0,¥). The graph is below,
left. The asymptote is x = 0 (the y-axis). The x-intercept is
approximately 0.12.
There is no y-intercept. The solution to g(x) = 1 is x = 0.2 accurate
to one decimal
place.
7.(a) The exact solution to 23-x = 52x+1 is
| x = |
3ln2-ln5 2ln5+ln2 | ; 0.12 |
(b) x = ± Ö3.
(c) x = 0.23, 0.50.
8. The point A is marked on the unit circle below, left. It is
in quadrant III.
The coordinates of A are (-0.65,-0.76) accurate to 2 decimal places.
9. The graph of y = -2cos(2x-p/6) appears above, right. The period is p, the amplitude is 2, and the phase shift is p/12 to the right.
10. At 14 years: $5,000 invested at 7% compounded continuously = 5000e(.07)(14) = $13,322.28. On the other hand, $4,000 invested at 9% compounded monthly = 4000(1+.09/12)(12)(14) = $14,035.54. Hence the friend's investment is worth more at 14 years.
11. At the same rate (computed to be ; 9.64%) the population will grow to 300,000 in 2017.
12. About 92.02 gms will remain at 150 years.
13. If q = arcsin(x), then
| tan(q) = |
x
|
14. The fires are approximately 10.3 miles apart.
15. The blimp travelled approximately 551 feet during the 3 minutes.