math problems with showing solutions

CU Boulder

MATH 1011 – College Algebra

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NAME:

Worksheet 3

The focus of this worksheet is further practice of the review skills from Sections 3.1-5.5. If you have not

already completed those problem sets in MyLab, you may find the content of those sections useful here.

Your regular problem sets in MyMathLab require only a correct answer to assess your mastery of the

skill. However, proper and complete communication of your mathematical techniques and practices are

also important skills. One of the goals of the worksheet exercises is to compel you to practice those written

communication skills. So, unless otherwise stated, in order to receive full credit, you must not only determine

the correct answer, you must clearly present correct relevant work leading to that correct answer.

A correct answer with insufficient or incorrect work may receive little or no credit. However, partial credit

can be earned even if the correct answer is not found, as long as the information that you have communicated

is clear, correct, and relevant.

You are welcome to discuss these problems with fellow classmates or tutors, but you must complete your

own write-up of the solutions. Identical (or nearly identical) solutions appearing on separate papers may

lose credit and may be considered a violation of the Honor Code.

Once you have completed your solutions, you must convert your document to a PDF and submit to Canvas.

Formats other than PDF and submissions made anywhere other than Canvas will not be accepted for grading.

1. (8 points) Suppose q is the rational function defined by q(x) =

5×2 + 19x − 4

.

x2 + 7x + 12

(a) Determine the domain of q(x) and write the answer in interval notation.

(b) Find the vertical asymptote(s), if any, of the graph of q(x). (Please makes sure to justify your

answer.)

CU Boulder

MATH 1011 – College Algebra

2

2. (8 points) Given that f (x) = 3×2 + 5x − 2, evaluate each of the following. Expand and simplify your

answer as much as possible and show all steps leading to your final answer.

(a) f (−2)

(b) f (a + 1)

(c)

f (x + h) − f (x)

h

3. (6 points) Suppose the point (−1, 5) is on the graph of a function f (x).

(a) If f is even, then what other point (besides (−1, 5)) must also lie on the graph of f (x)? Explain.

(b) If f is odd, then what other point (besides (−1, 5)) must also lie on the graph of f (x)? Explain.

CU Boulder

MATH 1011 – College Algebra

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4. (8 points) Let f (x) = −3

√

x

3

if − 4 ≤ x < −1
if − 1 ≤ x < 1
if x ≥ 1
(a) What is the domain of f (x)? Write your answer in interval notation. (Because the domain can
be read directly from the definition of the function, no work is required.)
(b) Determine the y-intercept of the function, if any. Make sure to justify your answer.
(c) Determine the x-intercept(s) of the function, if any. Make sure to justify your answer.
(Hint: Graphing the function may help, although a graph is not necessary. In any case, make
sure to fully justify your answer.)
CU Boulder
MATH 1011 - College Algebra
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5. (10 points) Let
f (x) = (x − 1)2 ,
g(x) =
1
,
x
h(x) =
1−x
.
1+x
Find (and fully simplify) the expression defining each function below. Additionally, find the domain
of the function.
f
(x)
(a)
g
(b) (h ◦ g)(x)
CU Boulder
MATH 1011 - College Algebra
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6. (10 points) Consider the parabola with equation f (x) = ax2 + bx + c shown in the graph here.
(a) What are the coordinates of the vertex of this
parabola?
(b) Does the parabola open upwards or downwards? What does this imply about the equation defining the parabola?
(c) What is the y-intercept of the graph of the
parabola?
(d) Determine the equation of the parabola. Write your final answer in the form f (x) = ax2 + bx + c.
CU Boulder
MATH 1011 - College Algebra
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7. (25 points) Solve each equation. For irrational solutions, round to 4 decimal places. In order to receive
full credit, you must show complete work arriving at your answer. Simply using some kind of “solver”
application and writing the answer down will receive no credit. However, it certainly makes sense to
use such technology to check your answers for accuracy.
(a) 220e0.037x = 440
(b) 5x = 86−x
CU Boulder
MATH 1011 - College Algebra
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(c) ex · e2x+1 = 95
(d) log6 (x2 − 15) = log6 (2x)
(e) ln(x) + ln(x + 8) = 3
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