MATH 22
FINAL EXAM
1. Find an equation of the line that passes through the point (2, −3)
which is parallel to the line 6x + 2y = 5 .
a) 3x + y = 3
SAMPLE A
6. A manufacturer cuts squares from the corners of a rectangular piece
of sheet metal that measures 5 in by 9 in. The metal is then folded
up to make an open-top box. Letting x represent the cut side-lengths
(in inches) of the squares, which of the following functions represents
Volume in terms of x?
b) x + 3y = −7
a) V = x3
c) x − 3y = 11
b) V = x(5 − x)(9 − x)
d) 3x − y = 9
c) V = (5 − 2x)(9 − 2x)
2. The point (5, 7) lies on a circle whose center is at (2, 3). Find an
equation of this circle.
7. Solve the inequality.
a) (x + 3)2 + (y + 2)2 = 5
2
d) V = x(5 − 2x)(9 − 2x)
3x3 + 10x2 − 8x ≤ 0
2
b) (x − 2) + (y − 3) = 25
c) (x + 2)2 + (y + 3)2 = 5
d) (x − 2)2 + (y − 3)2 = 25
3. Let f (x) = 2x2 + 4x + c where c > 2. The function f has how many
real zeros?
a) The function has no real zeros
a) (−∞, −4]
2
b)
− 4, 0 ∪
,∞
3
2
c)
− ∞, 0 ∪
,∞
3
2
d)
− ∞, −4 ∪ 0,
3
b) The function has ONE real zero
y
c) The function has TWO real zeros
d) More information is required
8. Which function is represented by the graph of y = f (x)?
-5 -4 -3 -2 -1
2
4. Find the solution to the quadratic inequality x − 10x < −24.
a) (0, 10)
b) (4, 6)
a) f (x) = (x − 3)(x2 + 1)(x + 4)
c) (−24, 0]
b) f (x) = −(x + 3)(x − 1)2 (x − 4)
d) (−∞, 4) ∪ (6, ∞)
c) f (x) = (x − 3)(x − 1)(x − 4)
d) f (x) = −(x + 3)(x2 + 1)(x − 4)
5. Find the range of the function.
√
f (x) = − x − 4 + 2
9. Solve the inequality.
3
≤2
x−6
a) (−∞, 4]
15
− ∞, 6 ∪
,∞
2
b) [4, ∞)
a)
c) (−∞, 2]
b) (−∞, 6)
15
c)
,∞
2
d) [2, ∞)
d) (−∞, 6) ∪ (6, ∞)
1
1
2
3
4
MATH 22
FINAL EXAM
SAMPLE A
10. Use polynomial long division to find the quotient Q(x) and remain- 13.
The function f has vertical asymptotes at x = −3 and x = 2, and
der R(x) when
a horizontal asymptote at y = 0. Which is the appropriate form for
4
3
2
(3x − 2x + 2) is divided by (x − 1).
f (x)?
a) Q(x) = 3x2 − 1,
2
b) Q(x) = −3x + x,
R(x) = −3
y
4
R(x) = x + 2
3
c) Q(x) = 3x2 − 2x + 3,
R(x) = −2x + 5
d) Q(x) = 3x2 − 2x + 1,
R(x) = −2x + 3
2
1
-5 -4 -3 -2 -1
-1
−7x
.
11. The one-to-one function f is defined by f (x) =
1 − 9x
−1
Find f , the inverse of f .
1
2
3
4
5
x
-2
-3
x
a) f −1 (x) =
9x − 7
b) f
−1
-4
1 − 9x
(x) =
−7x
c) f −1 (x) = −
d) f −1 (x) =
x
7 + 9x
1 + 9x
7x
a) f (x) =
a
(x − b)(x − c)
b) f (x) =
a(x − b)(x − c)
(x − d)(x − e)
c) f (x) =
a(x − b)
(x − c)(x − d)
a(x − b)
d) f (x) =
(x − c)
12. Consider the polynomial P (x) = 2x4 − 4x2 + kx − 4.
Find the value of k such that the remainder is 6 when P (x) is divided
by x − 2.
14. Find all asymptotes.
f (x) =
a) k = 5
(3x − 1)(2 − x)
(x + 2)(x − 7)
b) k = −3
a) y = 0,
c) k = 14
d) k = −21
b) y =
1
,
3
x = −2,
x = 7,
y = 2,
x = −2,
c) y = −3,
x = −2,
d) y = −3,
x = 2,
1
,
3
x=7
x=7
1
15. The expression (x2 + 2) 2 − x2 (x2 + 2)− 2
one of the following
expressions?
a) √
2
x2 + 2
p
b)
x2 + 2
3
c) (x2 + 2) 2 (1 − x2 )
1 − x2
d) √
x2 + 2
2
x=2
x=7
x = −2,
1
x=
is equivalent to which
MATH 22
FINAL EXAM
SAMPLE A
20. Which of the following is the graph of the function g(x) = 3 − ex ?
16. Solve for x.
log2 3 − log2 8 = log2 (x)
8
3
4
b) x = 6
3
3
c) x =
8
2
a) x =
y
1
a)
1
d) x =
6
-5 -4 -3 -2 -1
-1
1
2
3
4
5
1
2
3
4
5
x
-2
17. Given f (x) = 5x−3 , find f −1 (x).
-3
-4
a) f −1 (x) = log5 (x) + 3
b) f −1 (x) = ln(x) + 3
c) f
−1
y
4
ln(x) + 3
(x) =
ln 5
3
2
d) f −1 (x) = log5 (x − 3)
1
b)
18. Solve for x.
-5 -4 -3 -2 -1
-1
1
log36 x =
2
x
-2
1
a)
6
-3
-4
b) 18
c)
y
1
18
2
1
d) 6
c)
19. Solve for x.
5x
2
+5x+8
= 25−3x−8
-1
-1
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
x
-2
-3
-4
a) 4 and −4
-5
b) 8 and 3
-6
c) −8 and −3
y
d) −4
6
5
4
d)
3
2
1
-1
-1
-2
3
x
MATH 22
FINAL EXAM
21. Identify the false statement.
SAMPLE A
25. Find the domain of the function.
f (x) = − ln(3x + 2) + 4
√
1
a) ln( e) =
2
x
log x
b) log
=
y
log y
a) (−∞, 4]
2
b) (− , ∞)
3
c) log3 (20) is between 2 and 3
c) [4, ∞)
d) The range of f (x) = ln(x) is the set of all real numbers
2
d) (e 3 , ∞)
22. Solve for x.
log2 (2x2 − 4) = 5
r
a) x = ±
29
2
b) x = ±6
r
c) x = ±
2+
1
log2 (5)
2
√
d) x = ±3 2
FINAL EXAM - SAMPLE A - KEY
1. A 2. B 3. A 4. B 5. C 6. D 7. D 8. B 9. A 10. C 11. A 12. B
13. C 14. C 15. A 16. C 17. A 18. D 19. C 20. A 21. B 22. D 23.
A 24. A 25. B
23. Solve for x.
ln(x) + ln(x + 1) = ln(2x + 6)
a) x = 3
b) x = 3, −2
c) x = 5
d) no solution
24. Solve for x. Write the answer in terms of base-10 logarithms.
48x = 7x+9
a) x =
9 log(7)
8 log(4) − log(7)
b) x =
9 log(7)
8 log(4) − 1
c) x =
9 log(7)
7 log(4)
d) x =
log(7)
log(4)
4