Math 1090
Practice Test 1
6 February, 2012
1. Find the domain of each function:
√
(a) f (x) = x − 3
(b) g(x) =
1
x+2
2
(c) h(x) = x − 4
2. Solve the system of equations, if solutions exist.
(a)
1
y
3
= −x − 19
x + 31 y = 5
(b) 2x + y = 9
3x = y + 11
(c) y + 3z = −7
2x − y + 4z = −21
3y + 2z = 7
3. Given the graph of f (x) below, draw the graph of y = f (−x + 1) − 2.
5
2.5
-10
-7.5
-5
-2.5
0
2.5
5
7.5
10
-2.5
-5
4. Solve the linear inequality and graph the solution set on a number line:
2x + 4
x+5
−2<
+1
3
4
5. You are offered a job with two salary options: either (a) a fixed salary of $ 2000 per
month or (b) a salary of $ 1000 per month plus an 8% commission on sales. For what
range of slaes is plan (a) better?
6. Graph the solution set for this system of inequalities:
x − 3y < 16
2x + 3y < −4
4x − 3y > 10
7. Which of the following are functions?
(a) .
(b) (Ignore colors)
(c) (1, 0), (2, 4), (1, 3), (5, 7)
(d) (0, 4), (3, 6), (5, 8), (4, 9)
(e) x2 + y 2 = 1
(f) 3x + y = 6
8. Solve:
2(x − 7) = 5(x − 3) − x
9. Find the inverses of these functions.
(a) f (x) = 2x + 4
(b) g(x) =
(c) h(x) =
3x+1
x
√
3
3x + 1
10. Graph the solution to this inequality:
5x − 2y ≥ 3
11. Three less than four times a number is 25. What is the number?
12. Write the equation of each line described below.
(a) slope 45 , y-intercept −7
(b) slope
−2
,
3
passes through (4, 3)
(c) passes through (−4, 3) and (8, 10)
(d) passes through (2, 1), perpendicular to y = 32 x + 4
(e) passes through (8, 3), parallel to y = 14 x − 1
13. Find the maximum and minimum of the objective function y − 2x given the following
constraints:
x≥0
y≥0
1
y ≤− x+4
2
2x − 2y ≥ −6
(a) Draw the feasible region.
(b) Find the corners.
(c) Evaluate the objective function at each corner.
14. Find the slope of the line that passes through the points (14, 3) and (2, 12)
15. Find the slope and y-intercept of each line.
(a) y = 3x − 2
(b) 8x + 4y = 2
16. A craftsman sells leather belts for $15 each. If his tools required $300 in one-time startup costs and his variable costs are $9 per belt, how many belts must he sell to break
even?
17. Evaluate each function at the given value.
√
(a) f (x) = x2 + 3 at x = −1
(b) g(x) =
1
x−6
+ 2 at x = 4
18. Are lines with the following slopes parallel, perpendicular, or neither?
(a) m1 = − 21 , m2 = 2
(b) m1 = 3, m2 =
(c) m1 = 45 , m2 =
1
3
4
5
(d) m1 = −2, m2 = 2
19. The perimeter of a basketball court is 104 feet. The length is 15 feet more than the
width. What are the dimensions of the court?
20. Are these functions one-to-one? Find their inverses, if they exist.
(a) {(1, 4), (2, 6), (3, 6), (5, 7), (4, −2)}
(b) {(1, 6), (5, 7), (−1, 18), (6, 1)}
(c) y = x2
(d) y = 3x + 1
(e) (Don’t find the inverse)
3
2
1
-5
-4
-3
-2
-1
0
1
2
3
4
5
1
2
3
4
5
-1
-2
-3
(f) (Don’t find the inverse)
3
2
1
-5
-4
-3
-2
-1
0
-1
-2
-3
21. Describe the transformations necessary to take the graph y = x2 to y = 2(x − 3)2 + 1.
(Order matters!)
22. Let f (x) = x2 + 1 and g(x) = x + 3. Find the following:
(a) f + g(x)
(b) f − g(x)
(c) f g(4)
(d)
f
(−2)
g
(e) f ◦ g(x)
(f) g ◦ f (x)
23. Are these functions inverses?
4
x+2
4 − 2x
g(x) =
x
f (x) =
24. Sketch the graph of y = −|x + 1| + 3
25. Solve for x. Be sure to check your answer.
3 1
2 1
+ = +
x 4
3 x
26. Matt has two types of iodine in his lab. He has brand 206 iodine, which costs 28 cents
per ounce, and brand 822 iodine, which costs $1.70 per ounce. If Matt wants to make
367 ounces of iodine for a total cost of $203.26, how many ounces of each should he use?
27. Find functions f (x) and g(x) such that
f ◦ g(x) =
√
6x2 + 2