Download Practice Final Exam

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Math 100 Practice Final Exam (Seaver)
1.
(6 pts.) Solve the following
2 problems – (3 pts. each)
1
5
x 4
3
6
|3m – 1| = 5
3m – 1 = 5
3m – 1 = -5
1
5
6*( x  )  6*4
3
6
3m = 6
3m = - 4
2x + 5 = 24
x = 19/2
m=2
m = -4/3
2. (3 pts.) Solve (write in interval notation) and graph the inequality
-4p + 5  17
-5 -4
-3
-2
-1
0
3. (6 pts.) Given the equation 3x + 2y = 5, perform the following
a.
Find any x AND y intercepts
x
Y
0
5/2
5/3
0
5
-5
b. Find the slope of the line
y = - 3/2 x + 5/2
m = - 3/2
c.
1
Interval Notation: [-3, )
p  -3
Graph the equation (plot at least 3 points)
4. (3 pts.) Find the equation of the line (slope-intercept form) that goes through
points (2,3) and (-2,-5).
m = (-5-3) / (-2-2) = 2
y = 2x + b
3 = 2(2) + b
b = -1
y = 2x - 1
2
3
4
5
5.
(2 pts.) Are the lines y = 3x + 2 and -6x - 2y = -5 parallel, perpendicular, or neither?
-2y = 6x - 5
y = - 3x + 5/2 : neither
6. (4 pts.) Simplify the following (express using positive exponents)
2 problems – (2 pts. each)
2y2x-4
(3x2y3)3
27x6y9
(2y2) / (x4)
7. (2 pts.) Subtract (-3x2 + 10x + 3) – (-5x2 – 3x + 11) =
2x2 + 13x - 8
8. (3 pts.) Multiply and simplify (p + 3)(3p2 – 4p) =
3p3 + 5p2 – 12p
9. (6 pts.) Factor the following completely
2 problems – (3 pts. each)
x2 + 5x + 6
3m2 + 7m – 6
(x + 2)(x + 3)
10. (3 pts.) Divide (and simplify)
(3m - 2)(m + 3)
3x  3 x 2  6 x  7

x 1
x2 1
3( x  1) ( x  1)( x  1) 3( x  1)
*

x  1 ( x  7)( x  1) ( x  7)
2
6

x2 x7
11. (3 pts.) Simplify the complex fraction
4 x  13
2
x  9 x  14
2 x  14  6 x  12
2(4 x  13)
( x  2)( x  7)
( x  2)( x  7)

*
2
4 x  13
( x  2)( x  7)
4 x  13
( x  2)( x  7)
12. (3 pts.) Divide and simplify (2x3 – 6x2 – 4)  (x – 4)
4 | 2 -6 0 -4
8 8 32
2 2 8 28
13. (3 pts.) Solve
2x2 + 2x + 8 + (28)/(x-4)
x 5
 3
4 x
x2 + 20 = 12x
x2 – 12x + 20 = 0
(x-10)(x-2) = 0
x = 10 x = 2
14. (6 pts.) Find each root / simplify (assume all variables represent non-negative real numbers)
3 problems – (2 pts. each)
3
15. (2 pts.) Add and simplify
4x 
49x 8  7x4
 27  -3
2
3
2
 (2x)3 = 8x3
3 45x 3  x 5x 
9 x 5x  x 5x  10 x 5x
16. (3 pts.) Multiply and simplify ( 6  5)( 6  7) 
6  2 6  35  29  2 6
17. (2 pts.) Rationalize the denominator of
3
5
*
5
5

3
5
3 5
5
18. (4 pts.) Simplify and write each result in the form of a + bi
2 problems – (2 pts. each)
 50 
5i 2
(3 + 4i)2 =
9 +24i + 16i2 = -7 + 24i
19. (9 pts.) Solve the following quadratic equations (factoring, completing square, or quadratic formula)
3 problems – (3 pts. each)
x2 + 7x + 2 = 0
2 x  11x  3
 7  49  4(1)( 2)  7  41

2
2
2x
x2
5


x 1
x
x( x  1)
4x2 = 11x + 3
4x2 – 11x – 3 = 0
4x2 – 12x + x – 3 = 0
4x(x – 3) + (x – 3) = 0
(4x + 1)(x – 3) = 0
x = -1/4
x = 3
2x2 – (x2 + x – 2) = 5
x2 – x – 3 = 0
1  1  4(1)( 3) 1  13

2
2
20. (2 pts.) Graph the quadratic function f(x) = -2(x – 2)2 + 3
21. (3 pts.) Graph the quadratic inequality g(x) = -x2 – 2x + 8
g(x) = -(x2 – 2x + 1 – 1) + 8
g(x) = -(x – 1)2 + 9
22. (4 pts.) Solve the quadratic inequality (x - 2)(x + 1)(x + 5)  0
x = 2,-1,-5
-6
-5
-8(-5)(-1)
-40 < 0
-2
-4(-3)(3)
36 < 0
-1
0
2
-2(1)(5)
-10 < 0
3
1(4)(8)
32 < 0
(-,-5]  [-1,2]
23. (12 pts.) Given f(x) = x2 and g(x) = 3x + 2, find the following
4 problems – (3 pts. each)
(f – g)(x) =
(f * g)(x) =
x2 – 3x – 2
3x3 + 2x2
(g ◦ f)(x) =
(f(g(x))) =
3x2 + 2
(3x + 2)2 = 9x2 + 12x + 4
24. (6 pts.) Given f(x) = 2x – 2…
a. Find f -1(x)
b. Plot f(x) and f -1(x) on the same set of axes
x = 2y - 2
x + 2 = 2y
y = (1/2) x + 1
c. Show that the f -1(x) that you found in part a is indeed the inverse of f(x)
f(f-1(x)) = 2((1/2)x + 1) – 2
x+2–2
x
f-1(f(x)) = (1/2)(2x – 2) + 1
x–1+1
x