Precalculus Summer Refresher
1. Name the quadrant in which the point (–65, 1) is located.
A) I
B) II
C) III
D) IV
2. Find the distance between the points (–14, 4) and (–12, 5).
A)
65
B) √5
C) 757
D)
929
3. Find the midpoint of the segment joining the points (–11, 10) and (7, –4).
A) (–2, 3)
B) (–9, 7)
C) (3, –2)
D) (–4, 6)
4. Determine which point lies on the graph of the equation y = |3 – x| + 7.
A) (17, –13)
B) (–13, 17)
C) (–13, 23)
D) (0, 0)
2
5. What type of symmetry is represented by the graph of x = y − 10 ?
A)
B)
C)
D)
symmetry with respect to the x-axis
symmetry with respect to the y-axis
symmetry with respect to the origin
no symmetry
2
2
6. What type of symmetry is represented by the graph of 4 x + y = 81 ?
A) symmetry with respect to the x-axis
B) symmetry with respect to the y-axis
C) symmetry with respect to the origin
D) symmetry with respect to the x-axis, y-axis, and origin
7. The point (–9, 6) lies on a graph that is symmetric about the origin. Name the other
point that must also lie on the graph.
A) (6, –9)
B) (9, 6)
C) (–9, –6)
D) (9, –6)
8. Given the function f(x) = 5x – 2, evaluate f(x + 5).
A)
B)
C)
D)
5x + 23
5x – 23
5x2 + 23
5x + 3
9. Given the function H(x) = 1 – x2, evaluate H(x – 4).
A) –3 – x2
B) –3 + 8x – x2
C) –15 + 8x – x2
D) –3 – x
10. Given the function,
=√
−∞
∪
∞
A) (
, –9]
[9, )
B) ( −∞ , –9) ∪ (9, ∞ )
C) [–9, 9]
D) ( −∞ , –81] ∪ [81, ∞ )
− 81, state the domain in interval notation.
11. A projectile is fired straight up from an initial height of 210 feet, and its height is a
function of time, h(t) = –16t2 + 128t + 210 where h is the height in feet and t is the time
in second with t = 0 corresponding to the instant it launches. What is the height 4
seconds after launch?
12. Use the given graph to evaluate the function.
y = r(x)
a. r(–4)
b. r(3)
13. Use the vertical line test to determine if the graph below defines a function.
14. Determine if the function f(x) = x13 –x11 is even, odd, or neither even nor odd.
A) Even
B) Odd
C) neither even nor odd
15. Determine if the function h(x) = 2x2 + 7 is even, odd, or neither even nor odd.
A) Even
B) Odd
C) neither even nor odd
16. State (a) the domain, (b) the range, and (c) the x-intervals where the function is increasing,
decreasing, and constant. Find the values of f(0) and f(4).
17. Graph the piecewise-defined function. State the domain and range in interval notation. Determine
the intervals where the function is increasing, decreasing, or constant.
18.
The graph of y = |x| shifted up 20 and to the left 15. Write the resulting function.
A) y = |x + 15| + 20
B) y = |x – 15| – 20
C) y = |x – 20| – 15
D) y = |x + 20| + 1
19. The graph of y = |x| is reflected about the x-axis and shifted up 14. Write the
resulting function.
A) y = |x| + 14
B) y = –|x + 14|
C) y = –|x| + 14
D) y = –|x – 14|
2
20. Transform the function f(x) = x – 12 x + 30 to the form f(x) = c(x – h)2 + k, where c, h, and k are
constants, by completing the square.
21.
Given the functions f(x) = 2x + 2 and g(x) = –8x + 10, find (f + g)(x).
A) –6x + 12
B) –6x + 8
C) 10x + 12
D) –16x + 20
f ( x) =
22. Given the functions
A) 6 x − 4
2x + 7
B)
5 x 2 − 32
( x − 9)( x + 2)
C) 6 x 2 − 39 x + 52
( x − 9)( x + 2)
D) 6 x 2 − 39 x + 52
2x + 7
x+8
5x − 4
g ( x) =
x − 9 and
x + 2 , find (f + g)(x).
23. Given the functions f(x) = 3x2 + 8x and g(x) = –9x2 + 13x, find (f – g)(x).
A) 12x2 + 5x
B) 12x2 – 5x
C) –6x2 – 21x
D) –6x2 – 5x
⎛f ⎞
⎜ ⎟ ( x)
2
24. Given the functions f(x) = x – 25 and g(x) = x – 5, find ⎝ g ⎠
.
A) x + 5
B) x – 5
C) x2 – x – 20
D) x + 25
25. Given the functions f(x) = 3x + 4 and g(x) = 2x – 9, find ( g D f )( x ) .
A) 6x – 36
B) 6x – 5
C) 6x + 15
D) 6x – 1
26. Given the functions f(x) = 4x2 – 4 and g(x) = 4 – 3x, find ( f D g )(2) .
A) 12
B) –20
C) –24x2 – 32
D) 396
2
27. Given the functions f ( x ) = 8 x – 2 x – 3 , and g ( x) = –5 – 4 x , find ( f D g )( x ) and ( g D f )( x )
28.
Determine if the relationship f = {(–2, 18), (–17, –5), (14, –5), (–14, 2)} is a one-to-one
function.
A) not a function
B) a one-to-one function
C) a function, but not one-to-one
29.
Determine if the relationship x = (y – 2)2 + 17 is a function. If it is a function, determine
if it is a one-to-one function.
A) not a function
B) a one-to-one function
C) a function, but not one-to-one
30. The function f(x) = x2 + 7, x ≥ 0 is a one-to-one function. Find its inverse.
A)y = x + 7
B y = x−7
)
C y = x+7
)
D
1
) y = x2 + 7
31. The function y = 6x-5 is a one-to-one function. Find its inverse.
A)
1
5
y =− x+
6
6
B)
1
5
y = x+
6
6
C)
1
y=
5 − 6x
D)
1 1
y= −
5 6x
32. Determine whether the function is a one-to-one function.
33. Given the graph of a one-to-one function: plot its inverse.
34. Given the graph of a one-to-one function: plot its inverse.
35. Find the slope-intercept form of the line through the points (3, - 5) and (2, - 1).
36. Find the equation of a line parallel to
= 7 − 10 that passes through (2, - 4).
37. Find the equation of a line perpendicular to
38. Graph
,
− 4,
<3
×≥ 3
= 3 − 6 that passes through (9, 5).
39. Graph 2 + 3 < 4
40. Solve this system of equations algebraically.
−2 +
= −1
2 + 3 − 2 = −3
+ 3 − 2 = −2
41. Use matrices to solve this system.
+
=4
2 + 2 + 4 = 10
+6 +8 =4
42. Solve 14
+ 19 − 3 = 0 by factoring.
43. Solve 2
+ 11 − 21 = 0 using completing the square.
44. Solve 6
+ 7 + 2 = 0 using the quadratic formula.
45. Solve 6
+ 8 + 5 = 0 using the quadratic formula.