Algebra problems

1. Select the correct description of right-hand and left-hand behavior of the graph of the polynomial function.

f (x) = (1/5) x³ + 5x

Rises to the left, falls to the right
Rises to the right, rises to the left
Falls to the left, rises to the right
Falls to the right
Falls to the left, falls to the right

2. Select the correct description of right-hand and left-hand behavior of the graph of the polynomial function.

f(x) = 4x² - 5x + 4

Falls to the left, rises to the right
Falls to the left, falls to the right
Rises to the left, rises to the right
Rises to the left, falls to the right
Falls to the left

3. Find all the real zeroes of the polynomial function.

f(x) = x² - 25

-25
5
-5
25
±5

4. Use synthetic division to divide.

(4x³ + x² - 11x + 6) ÷ (x + 2)

4x² - 5x - 6
4x² - 7x + 3
4x² - 2x - 2
4x² + 5x - 12
4x² + 7x - 4

5. Use the Remainder Theorem and synthetic division to find the function value. Verify your answers using another method.

h(x) = x³ - 6x² - 5x + 7

h(-8)

-849
-847
-851
-848
-845

6. Find all the rational zeroes of the function.

x³ - 12x² + 41x - 42

-2, -3, -7
2, 3, 7
2, -3, 7
-2, 3, 7
-2, 3, -7

7. The total revenue R earned (in thousands of dollars) from manufacturing handheld video games is given by

R(p) = -25p² + 1700p

where p is the price per unit (in dollars).

Find the unit price that will yield a maximum revenue.

$38
$35
$36
$37
$34

8. Find the domain of the function f (x) = 8x² / (x² - 49)

Domain: all real numbers x except x = 7
Domain: all real numbers x except x = ±49
Domain: all real numbers x except x = ±8
Domain: all real numbers x except x = -7
Domain: all real numbers x except x = ±7

9. Find the domain of the function and identify any vertical and horizontal asymptotes.

f (x) = 2 / (x - 5)³

Domain: all real numbers x
Vertical asymptote: x = 0
Horizontal asymptote: y = 0

Domain: all real numbers x except x = 2
Vertical asymptote: x = 0
Horizontal asymptote: y = 0

Domain: all real numbers x except x = 5
Vertical asymptote: x = 0
Horizontal asymptote: y = 2

Domain: all real numbers x
Vertical asymptote: x = 0
Horizontal asymptote: y = 2

Domain: all real numbers x except x = 5
Vertical asymptote: x = 5
Horizontal asymptote: y = 0

10. Simplify f and find any vertical asymptotes of f.

f (x) = {x² (x + 3)} / (x² + 3x)

x+3; vertical asymptote: x = -3
x; vertical asymptote: none
x; vertical asymptote: x = -3
x-3; vertical asymptote: none
x2; vertical asymptote: none

11. Determine the equations of any horizontal and vertical asymptotes of

f (x) = (x² - 1) / (x² + 4x - 5)

horizontal: y = 5; vertical: x = 0
horizontal: y = 1; vertical: x = -5
horizontal: y = 1; vertical: x = 1 and x = -5
horizontal: y = -1; vertical: x = -5
horizontal: y = 0; vertical: none

12. Identify all intercepts of the following function.

g (x) = (x² + 3) / x

x-intercepts: (±3, 0)
no intercepts
x-intercepts: (-3,0)
x-intercepts: (0,0)
x-intercepts: (3,0)

13. Select the correct graph of the function.

f (x) = 8 / (x - 5)

14. The game commission introduces 100 deer into newly acquired state game lands. The population N of the herd is modeled by

N = [{30 (5 + 5t)} / (1 + 0.04t)], t ≥ 0

where t is the time in years. Find the populations when t=40. (Round your answer to the nearest whole number.)

1,442 deer
1,632 deer
1,594 deer
1,550 deer
1,500 deer

15. Evaluate the function at the indicated value of x. Round your result to three decimal places.

Function: f(x) = 6000(6x) Value: x = -1.3

584.191
784.191
-584.191
684.191
-784.191

16. Select the graph of the function.

f (x) = (1/3^)-x

17. Use the One-to-One Property to solve the equation for x.

e^x2-6 = e^5x

x = -6
x = 5
x = 6, -1
x = -6, -1
x = -6,1

18. log₃₆6 = 1/2

36^½ = -6
36^½ = 6
6^½ = 36
36^½ = -1/6
36^½ = 1/6

19. Write the exponential equation in logarithmic form.

27² = 729

log₂₇729 = 2
log₂₇729 = 1/2
log₇₂₉27 = 2
log₂₇729 = -2
log₂₇2 = 729

20. Find the exact value of the logarighmic expression without using a calculator.

4 ln e⁷

7
28
4
e
1

21. Condense the expression to the logarithm of a single quantity.

ln₃10 + ln₃x

ln₃(10 - x)
ln₃10/x
ln₃(10 + x)
ln₃10x
ln₃10x

22. Solve for x.

6^x = 1,296

6
10
4
-6
-4

23. Solve the exponential equation algebraically. Approximate the result to three decimal places.

e^x - 8 = 12

ln20 ≈ 2.485
ln20 ≈ 2.996
ln20 ≈ -2.485
ln20 ≈ 2.079
ln20 ≈ -2.996

24. An initial investment of $9000 grows at an annual interest rate of 5% compounded continuously. How long will it take to double the investment?

1 year
14.40 years
13.86 years
14.86 years
13.40 years

25. The populations (in thousands) of Pittsburgh, Pennsylvania from 2000 through 2007 can be modele by p = 2632 / (1 + 0.083 e^0.0500t) where t represents the year, with t = 0 corresponding to 2000. Use the model to find the population in the year 2001.

2,418,774
2,419,774
2,421,774
2,420,774
2,422,774