MATH 1050 SECTION 3
FALL 2010
PRACTICE EXAM I
1. General Guidelines
1.1. Calculators. You will not be permitted to use calculators. As such, the problems will involve
relatively simple arithmetic that you should be able to do by hand.
1.2. Material Covered. The exam will cover material through section 2.1 in the book. You should
know how to do all questions from the homework.
1.3. Showing Work. You should show enough work so that I can follow your thought process. When
in doubt, show more detail. I also highly recommend that you check your answer for consistency
where possible. For example, if you are asked to find the equation of a line passing through two
points, you should check that the two points satisfy the equation you’ve obtained.
1.4. Studying. Mathematics is a set of skills than a collection of facts. Practice is the key to
mastering these skills. I recommend that you work as many problems as possible rather than simply
memorizing formulas from the book. Work the problems below and then rework as many homework
problems as you can. If you have time, work extra problems from the book.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
2. Practice Problems
p
Simplify the expression
54xy 4 . Make sure to put in absolute value bars where they are
√
2
needed. Answer: 3y 6x
Multiply out (x + 3)(x + 4) and simplify. For which values of x should this expression go to 0?
Plug these values in to the simplified expression to verify your work. Answer: x2 + 7x + 12,
with roots −3 and −4.
Factor x2 + 10x + 25. Hint: perfect square. For which values x should this polynomial go to
0? Verify your work by plugging these values in to the original expression. Answer: (x + 5)2 .
The only root is x = −5 with multiplicity 2.
Factor 12x2 − 48. Hint: difference of two squares. Make sure to pull out integer factors.
Answer: 12(x − 2)(x + 2).
Factor 4x(2x − 1) + (2x − 1)2 by grouping. Make sure that each of your linear factors (with
form ax + b) is completely simplified. Answer: (2x − 1)(6x − 1).
Factor 2x2 + 9x + 9. Answer: (2x + 3)(x + 3).
Use the quadratic formula to solve the equation
√
x2 − 10x + 22 = 0.
Answer: 5 ± 3
(8) Start with the expression
x2 + 8x − 20
.
x2 + 11x + 10
Factor the numerator and denominator completely. Simplify by cancelling common factors.
What point must be removed from the domain of the new expression to account for the
(x − 2)
cancelled factors? Answer:
, x 6= −10
(x + 1)
1
5
(9) Find all solutions to the equation |3x + 2| = 7. Answer: x = , −3.
3
2
1
2
(10) Find all solutions to the equation
=
+
. Take special care to show
(x − 4)(x − 2)
x−4 x−2
your thought process in this question. Answer: No solution.
(11) Find the solution sets to the inequalities |x + 4| > 2 and |x − 3| < 4. Answer: My original
explanation of this problem in class was confusing. First of all, we are considering the
solutions to these problems separately. For details on how to solve problems like these, I
suggest you see the appendix to the book, page A64. For the inequality |x + 4| > 2, the
solution breaks into two parts:
x+4>2
and
−(x + 4) > 2.
We solve each of these separately. The solutions are x > −2 and x < −6. The inequality
|x − 3| < 4 turns into
−4 < x − 3 < 4.
The solutions is then
−1 < x < 7.
(12) Find the distance between the points (3, 11) and
√ (2, 4). Find the midpoint of the line segment
connecting these points. Answer: Distance: 5 2. Midpoints: (5/2, 15/2).
(13) Write an equation for the circle with center at (2, 3) and radius 3. Graph this circle. Answer:
(x − 2)2 + (y − 3)2 = 9.
(14) Find the equation of the line passing through (4, 6) and (9, 2). Find the slope of this line.
Using the equation you wrote down, find the y-intercept for this line. Use your calculations
to write the slope intercept form of the equation of this line. Verify both equations (the
two point form and the point-slope form) by checking that the points satisfy the equations.
Answer:
4
(y − 6) = − (x − 4).
5
There are other possible correct answers for the first part.
4
46
y =− x+ .
5
5
(15) Return to the graph of the circle above. Does this graph come from a function? Hint: vertical
line test. Answer: No.
(16) What does it mean for a function to be one to one? How do we determine if a graph is one
to one? Answer: At most one input gives any specified output. Vertical line test.
(17) Does the following equation represent y as a function of x?
|y| = 4 − x.
Answer: No. For example, if x = 3, y can be plus or minus 1.
(18) Find the average rate of change of the function x2 + 3x + 4 from x = 3 to x = 7. Sketch the
function. Draw a line connecting the points on the graph at x = 3 and x = 7. How does this
line relate to the average rate of change? Answer: 13.
(19) Is the function f (x) = x6 − 2x2 + 3 even or odd? What symmetry does the graph have?
Answer: Even. y-axis symmetry.
(20) Start with the function f (x) = x3 . Move it six units to the left, six units downward and
reflect across the y-axis. Write down the expression for the new function h(x). Answer:
h(x) = (−x + 6)3 − 6.
2
(21) Let g(x) = |3 − x| − 8. What is the basic function type? Write down the series of geometric
transformations that takes you from the basic function to g(x). Check your work by applying
these transformations to the basic function to get g(x).Answer: Start with |x|. Shift 3 units
to the left: |x + 3|. Reflect across the y axis: |−x + 3|. Shift down 8 units: |−x + 3| − 8.
(22) Let g(x) = 5x2 and let h(x) = x3 +4. Compute g ◦h(x) and h◦g(x). Compute g −h and g +h.
Do not simplify. Answer: g◦h(x) = 5(x3 +4)2 . h◦g(x) = (5x2 )3 +4. (g−h)(x) = 5x2 −(x3 −4).
(h − g)(x) = x3 + 4 − 5x2 .
p
f
(23) Let f (x) = 24 − x2 and let g(x) = x−4. Compute . What is the domain of the quotient?
g
√
√
√
24 − x2
f
=
. Domain: − 24 ≤ x ≤ 24 with x 6= 4.
Answer:
g
x−4
3
2
(24) Let f (x) = x − 5x − 14x. Evaluate f (−2). Answer: 0.
(25) Plot y = x2 + 2x + 3. Find the vertex and label it on your graph. Evaluate y for at least 2
other values of x and label these points. Answer: Vertex: (−1, 2)
−4x − 3
2x − 3
. Answer:
(26) Find the inverse of f (x) =
3x
+
4
3x − 2
√
(27) f (x) = x + 2 − 1. What is the domain of f ? Draw a graph of f . Answer: x ≥ −2.
(28) Find the equation of a line (in slope-intercept form) that is perpendicular to the line 2x−3y =
3
6 and has the same y-intercept. Answer: y = − x − 2.
2
(29) Find the equation of a parabola with roots −2 and 3 and with y-intercept (0, 24). Answer:
y = −4(x + 2)(x − 3).
(30) Let f (x) = 2x + x2 and g(x) = x + 1. Find f (g(x)). Answer: f (g(x)) = 2(x + 1) + (x + 1)2 .
(31) Find the equation of a circle with (2, 4) and (−3, −4) as the endpoints of a diameter. Answer:
89
1
(x + )2 + y 2 =
2
4
√
(32) Find the domain of the function f (x) = 2 x + 3. x ≥ −3.
(33) Let f (x) = 4x − 1 and g(x) = x2 + x. Find (f ◦ g)(−2). Answer: 7.
(34) Find the slope-intercept equation of the line through (2, 3) and perpendicular to the line
1
y = 2x − 3. Answer: y = − x + 4.
2
(35) Find the vertex of the parabola defined by x2 − 16x + 100. Answer: (8, 36).
(36) Find the equation of a parabola with roots −2 and 3 and with y-intercept (0, 24).
2x2 + x − 6
(37) h(x) = 2
.
6x − 5x − 6
Factor the numerator and denominator of h. Simplify h, noting any domain restrictions.
(2x − 3)(x + 2) x + 2
3
Answer:
,
, x 6= .
(2x − 3)(3x + 2) 3x + 2
2
(38) Plot y = x2 + 4x + 3. Label at least three points on your graph.
3x − 2
−3x − 2
(39) Find the inverse of
. Answer:
.
4x
+
3
4x − 3
√
(40) f (x) = x − 1 + 2. What is the domain of f ? Draw a graph of f , labelling at least two
points on your graph. Answer: x ≥ 1.
(41) Find the equation of a line (in slope-intercept form) that is perpendicular to the line 3x−2y =
2
12 and has the same y-intercept. Answer: y = − x − 6.
3
(42) Find the equation of a parabola with roots 2 and −3 and with y-intercept (0, −12). Answer:
y = 2(x − 2)(x + 3)
2
(43) Find an equation of the line passing through (2, 3) and (5, 1). Answer: (y − 3) = − (x − 2).
3
3x − 4
(44) Let f (x) =
.
2x + 1
3
(45)
(46)
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
(55)
What is the domain of f ? Find f (3). If g(x) = 3x2 + 1, what is f ◦ g(x)? Do not
1
5
3(3x2 + 1) − 4
simplify. Find f −1 (x). Answer: Domain: x 6= − . f (3) = . f ◦ g =
.
2
7
2(3x2 + 1) + 1
x+4
f −1 (x) =
.
3 − 2x
Describe how to obtain the graph of f (x) = (x − 1)2 + 2 from the graph of g(x) = x2 by a
series of suitable shifts. Graph f and g. Answer: Shift 1 unit to the right and then 2 units
up.
x−1
. What is the x-intercept? Answer: (1, 0).
Let f (x) = 2
x − 3x
−2
4 2
Simplify 3x (2x ) . Answer: 12x6 .
Factor 9x2 + 10x + 1. Answer: (9x + 1)(x + 1).
Perform the division and simplify:
(x + 6)(x + 1)
x2 − 36 x3 − 6x2
.
÷ 2
. Answer:
x
x +x
x2
Perform the multiplication and simplify:
4y − 16 2y + 6
8
·
. Answer: − .
5y + 15 4 − y
5
Perform the addition or subtraction and simplify:
(a)
5
6−
x+3
6x + 13
.
Answer:
x+3
(b)
1
x
−
x2 − x − 2 x2 − 5x + 6
−3 − x2
Answer:
.
(x − 2)(x + 1)(x − 3)
(c)
2
1
1
+
− + 2
x x + 1 x3 + x
2−x
Answer: 2
.
x +1
Solve the equation and check your solution. If there is no solution, explain.
1
1
10
+
= 2
x−3 x+3
x −9
Answer: x = 5.
Determine whether the equations represents y as a function of x:
(a) x2 + y 2 = 4. Answer: No.
(b) x2 + y = 4. Answer: Yes.
Find two functions f (x) and g(x) such that f ◦ g(x) = h(x), where h(x) = (2x + 1)2 . Answer:
There is more than one correct answer, but one possibility is f (x) = x2 and g(x) = 2x + 1.
Find the inverse of f (x) = (x + 2)3 . Answer: f −1 (x) = x1/3 − 2.
4