Mathematics 120 (Precalculus) Fall, 2007

Review: Examination I

prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, NY 11451

Email: malkevitch@york.cuny.edu

web page: http://www.york.cuny.edu/~malk

1. Determine the value(s) of x for which the following equations and inequalities are satisfied. (If the solution set is an interval or ray, write down the solution in interval notation.)

a. x = 7

b. -x = 3

c. 2x = 16

d. 9x = 36

e. -x = 13

f. -5x = 35

g. 13x = 15

h. x + 11 = -2

i. 3x - 5 = 7

j. 2x + 12 = -6 + 3x

k. 5(x - 2) = 2x - 40

l. x > 12

m. 2x -8 < x + 4

n. -x + 3 > -2x + 12

o. 2(x - 9) > -3x +5

p. -(x - 9) < x -12

q. | x + 4 | ² 7

r. | x - 3 | > 5

s. -2x + 6 > -5

t. 2x - 6 > -5

u. | -x + 3 | = 4

v. -x + 6 > -(2x +5) and 4x > x +2

w. | x | = 9

x. | x - 4 | = 7

2. Sketch the graph of the following equations: (If the equation represents a line show, if possible, the x and y intercepts; for circles show the center (and compute the length of the radius) and 4 points on the circle.)

a. x = 5

b. y = -3

c. x + 2 y = 8

d. y = -x + 9

e. x2 + y2 = 16

f. x2 + y2 - 8x = 0

g. x2 + y2 + 4y - 6x = -2

h. (x - 4)2 + (y + 6)2 = (1/4)2

i. x/3 + y/7 = 1

j. -x + 3y = 9

k. 2x2 + 12 x + 2y2 - 8y = 3

3. Write down the following using interval notation:

a. -1 < x

b. x > 3

c. -2 < x < 9

d. x ³ 4

e. -2 < x ² 5

f. 3 ² x ² 9

4. Write down using inequalities the points which are represented by the following intervals (and draw a sketch of the interval:

a. (-2, 1)

b. (-6, -2 ]

c. [ 6, 8 ]

d. (-°, 5)

e. (6, ° )

5. Find the center and radius of the following circles and be prepared to draw a sketch of the circle which shows as least 4 points on the circle.

a. x2 + y2 = 100

b. x2 + y2 - 4x + 8y = -4

c. x2 - 6y + y2 + 10x = +2

d. x2 - y2 + 6y = -2y2

6. Given the function defined by the formula y = f(x) = -x2 + 4.

(i) Compute:

a. f(3)

b. f(-2)

c. f(0)

d. f(-1/2)

e. f(a)

f. f(a +1)

g. f(x +h) - f(x)

ii.

What are the domain and range of f(x).

7. Given the function defined by the formula y = g(x) = -

a. g(3)

b. g(-2)

c. g(0)

d. g(-1/2)

e. g(a)

f. g(a +1)

g. g(x +h) - g(x)

ii. What are the domain and range of g(x).

8. Given the function defined by the formula y = h(x) = x/(x-2)(x-4)

a. h(3)

b. h(-2)

c. h(0)

d. h(-1/2)

e. h(a)

f. h(a +1)

g. h(x +h) - h(x)

ii. What are the domain of h(x).

9. Sketch the location of the points (-2, -3), (-3, 2), (-4,2), (4, 5).

10. Given triangle O = (0, 0), A = (-3, -4) and B = (5, 12), find the lengths of the sides of the triangle and its perimeter. Write down the equations of the sides of this triangle.

11. Are the functions f(x) = 3x with domain the set of integers and the function g(x) = 3x with domain the real numbers the same?

12. Are the functions g(x) = | x - 4 |, domain all real numbers and h(x) = | 4 -x | domain all real numbers the same function?

13. For the function f(x) = 2x2 compute the value of f(x + 3) - f(3).

14. For the function f(x) = 9x compute the value of f (x +4) - f(4).

15. Be able to the use the vertical and horizontal line tests to determine the domain and range of a function which is defined with the use of a diagram showing the graph of the function.

15.

a. Write down the equation of the line given the points:

i. (0, 3) and (-4, 5)

ii. (0, 3) and (0, -11)

iii. (-2, -5) and (-1, -3)

iv. (-2, 0) and (2, 7)

b. Find the slope (if defined), intercepts, and two points on the lines below:

i. x + 3y = 7

ii. -3y = 7

iii. 2x + 5y = 11

iv. -5 x = 7

v. -3x + 6y = -5

16. Find the equation of a line parallel to 2x + 5y = 7 and which goes through the point (1,2).

17. Find the equation of a line perpendicular to 2x + 5y = 7 and goes through the point (1, 2).

18. Sketch a graph of the following equations:

a. y = 2x2

b. y = -2x2

c. y = 2x2 + 4

d. y = 2(x+3)2

e. y = -2(x-5)2 + 4