Math 117 Practice Problems for Test I
4. Find the range of the function f(x) = 2(x – 3)6 – 8
5. Find the range of the function y = 9 – 2x4 – 5x2
6. Find the range of the function y = 3x2 + 1492
7. Expand and simplify fully:
(a + b) ( a – c) – (a – b) (a + c) + (a + c) (b – c)
8. Solve for x:
3(1 – 2(1 – 4x)) = 10x – (3 – 5x) + 45
9.
Simplify by removing brackets:
a – 2b – [4a – 6b – {3a – c + (5a – 2b – (3a – c + 2b))}]
10. Solve for x in each equation below:
(a)
84 + (x + 4)(x – 3)(x + 5) = (x + 1)(x + 2) (x + 3)
(b)
3(x – 1)2 – 3(x2 – 1) = x – 15
11. Solve the following equation for x:
(2x – 1)(3x + 5) + 7x + 5 = (6x + 5)(x – 3) – 10(x + 4)
12. True or False? (You need not provide an explanation.)
(A) (a + b + c)2 = a2 + b2 + c2 for all real numbers a, b and c.
________
(B)
a  b  a  b for all a  0, b  0 ________
(C)
0  0 _______
(D)
a

b
(E)
550 = 1
(F)
00 = 1 __________
(G)
a
b
for all a  0, b  0 ________
__________
5x 3  x  1
x 1
 5 3
3
x
x
13. Find all values of x satisfying the equation:
2x  1 4x  7

4 x  1 8x  3
14. Find the largest value of f(x) = 5 – (3x4 + 1)2016
15. Compute without using a calculator: 75002 – 25002
16. Factor fully each of the following:
(a) (a – 2b + 3c)2 – (a + 4b – 7 d)2
(b) x8 – 256
(c) 6x2 – 19x – 42
(d) 10x2 + 23xy – 5y2
17. Factor each of the following polynomials:
(a) 20x2 + 3x – 9
(b) 40x2 – 63x + 18
18. Let f(x) = 1/(x+1)2, and let g(x) = 1 + 4x2.
(a) Calculate
f ( x  h)  f ( x )
. Simplify your answer as much as possible
h
(b) Calculate
g ( z  k )  g ( z)
. Simplify your answer as much as possible.
k
19. Factor fully each of the following:
(a) 2x5 – x4 – 15x3
(b) 12x2 +17xy – 99y2
(c) x2 – 16y2
(d) 256 a8 – b16
20. Find the domain of the function:
g ( x)  2016  x 99 ( x  5)( x  3)( x  4)
21. Find an equation of the inverse of the function. That is, solve for x as a
function of y.
y
x7
1  3x
22. Find all values of x satisfying the equation:
3x  11 24 x  9

x 1
8x  3
23. Simplify fully the following expression:
x  1 x2  2

x  3 x2  9
24.
Let y  f ( x)  1 
1
.
3
x
(a) Evaluate f -1(2). (This means: solve for x when y is 2.)
(b) Evaluate (f(2))-1.
(c) Evaluate f(2-1).
25.
The quantity, Q mg, of nicotine in the body t minutes after a cigarette is
smoked is given by Q = g(t). Using a complete sentence, interpret the statement
g(20) = 0.36 without using any mathematical terminology.
26. Consider the following piecewise defined function:
 x  4 if x  2

f ( x)  17 if x  2
 x 2  1 if x  2

Find the value of each of the following: G(1), G(2), G(-1), G(-3)
26. Let G ( x)  1 
x
.
x7
27. Simplify the following complex fraction:
2
1  x2
1
1

1 x 1 x
28. Simplify the following complex fraction:
a
x
 2
2
x
a
1
1
1
  2
2
a
ax x
29. Simplify the following complex fraction:
a  2b a  2b

a  2b a  2b
a 2  4b 2
1
(a  2b) 2
30. Find the inverse of the function
g ( x) 
5
1  x3
31. Let g(x) = 3(x2 + 1)(x – 4). Compute (without simplifying)
(a) g(0)
(b) g(2x)
(c) g( x + 4)
(d) g(x – 1)
(e) g(2/x)
32.
Let f ( x) 
x
.
x 1
Compute and simplify
f ( x )  f ( x  h)
.
h
33. Simplify fully:
xa
xa

2
2
x  a ( x  a) 2
34. Simplify fully the following complex fraction:
1
1 x
1
1 x
x
35. Factor and simplify each of the following:
(a)
(a + 2b)2 – 16x2
(b)
(2x + a – 3)2 – (3 – 2x)2
(c)
8pq3 + 32pqr – 192 p2q5r
36. Find the domain of each of the following functions:
37. Suppose that f and g are functions completely defined by the rules below.
(c)
x
f(x)
x
g(x)
0
17
0
19
3
5
2
11
4
7
4
10
7
4
7
13
11
7
10
9
13
4
14
0
19
33
a
Find f (0)
b
Find g  f (7)
b
Find f g (7)
38. Find the range of each of the following functions:
(a) y = 1 + x3
(b) y = (x – 2)4 + 123
(c) y = 9 – (x – 4)2
(d) y = 7 + x4
39. Compute without using a calculator: 20202 – 20102
40. Let g ( x) 
x
.
x3
(a) Find and simplify (g (2))-1.
(b) Find and simplify g(2-1)
(c) Find and simplify g-1(2)
41. Which of the following functions, if any, is one-to-one? (Identify any and all
that are one-to-one.) You need not give reasons for your answers.
(a) y = (x – 1) (x – 3)
(b) y = x8 + x2 + 5
(c) y = (x + 1)2
(d) y = (x + 3)3 + 1
(e) y = 5 + 4x1/5
42. Let f ( x)  1 
x
.
x2
Find f -1(3).
43. Let G(x) = x3 + 5. Compute and simplify:
G (1  h)  G (h)
h
I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to
study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce and
agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry,
and porcelain.
- Letter from John Adams to Abigail Adams, May 12, 1780