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    1. QUESTION

    The quiz is worth 100 points. There are 9 problems. This quiz is open book and open notes. This means that you may refer to your textbook, notes, and online classroom materials, but you must work independently and may not consult anyone (and confirm this with your submission). You may take as much time as you wish, provided you turn in your quiz no later than Sunday, Nov 11.
    Show work/explanation where indicated. Answers without any work may earn little, if any, credit. You may type or write your work in your copy of the quiz, or if you prefer, create a document containing your work. Scanned work is acceptable also. In your document, be sure to include your name and the assertion of independence of work.
    General quiz tips and instructions for submitting work are posted in the Quizzes module.
    If you have any questions, please contact me by e-mail.

    1. (4 pts) Which of these graphs of relations describe y as a function of x?
    That is, which are graphs of functions? Answer(s): ____________
    (no explanation required.) (There may be more than one graph which represents a function.)

    (A)
    (B)

    (C)
    (D)

    2. (10 pts) Consider the points (–5, –6) and (–1, 10).
    (a) State the midpoint of the line segment with the given endpoints. (No work required)

    (b) If the point you found in (a) is the center of a circle, and the other two points are points on the circle, find the length of the radius of the circle. (That is, find the distance between the center point and a point on the circle.) Find the exact answer and simplify as much as possible. Show work.

    3. (16 pts) Consider the following graph of y = f (x).

    (no explanations required)

    (a) State the x-intercept(s).

    (b) State the y-intercept(s).

    (c) State the domain.

    (d) State the range.

    4. (9 pts) Let f(x) = (x + 5)/(x- 8)^2

    (a) Calculate f(-2). (work optional)

    (b) State the domain of the function f(x) = (x + 5)/(x- 8)^2

    (c) Find f(a+7) and simplify as much as possible. Show work.

    5. (6 pts) Given f(x)=x-5 and g(x)=|x+7|, which of the following is the domain of the quotient function ? Explain. 6._______
    A. (-7,∞)
    B. (-∞,-7) ∪ (-7,5) ∪ (5,∞)
    C. (-7,5) ∪ (5,∞)
    D. (-∞,-7)∪(-7,∞)

    6 (8 pts) For income x (in dollars), a particular state’s income tax T (in dollars) is given by

    T(x)={■(0.028x&if 0 ≤x≤2,[email protected]+0.035(x-2500)&if 2,500<x≤7,[email protected]+0.050(x-7,500)&if x=”” style=”box-sizing: border-box; color: rgb(51, 51, 51); font-family: Roboto, -apple-system, BlinkMacSystemFont, “Segoe UI”, Roboto, “Helvetica Neue”, Arial, sans-serif, “Apple Color Emoji”, “Segoe UI Emoji”, “Segoe UI Symbol”; font-size: 13px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial;”>7,500)┤

    (a) What is the tax on an income of $6,300? Show some work.

    (b) What is the tax on an income of $63,000? Show some work.

    7. (20 pts) Let y = 2 – 2×2.

    (a) Find the x-intercept(s) of the graph of the equation, if any exist. (work optional)

    (b) Find the y-intercept(s) of the graph of the equation, if any exist. (work optional)

    (c) Create a table of sample points on the graph of the equation (include at least six points), and create a graph of the equation. (You may use the grid shown below, hand-draw and scan, or you may use the free Desmos graphing calculator described under Course Resource to generate a graph, save as a jpg and attach.)

    x y (x, y)

    (d) Is the graph symmetric with respect to the y-axis? _____ (yes or no). If no, state a point on the graph and state the appropriate reflection point which fails to be on the graph, as done in section 1.2 homework in the textbook.

    (e) Is the graph symmetric with respect to the x-axis? _____ (yes or no). If no, state a point on the graph and state the appropriate reflection point which fails to be on the graph, as done in section 1.2 homework in the textbook.

    (f) Is the graph symmetric with respect to the origin? _____ (yes or no). If no, state a point on the graph and state the appropriate reflection point which fails to be on the graph, as done in section 1.2 homework in the textbook.

    8. (12 pts) Let f (x) = 4×2 + 2x – 8 and g(x) = 1 – 2x.

    (a) Evaluate the function g – f for x = –3. That is, find (g – f)(–3). Show work.

    (b) Evaluate the function fg for x = –3. That is, find (f g)( –3). Show work.

    (c) Find the difference function (f – g)(x) and simplify the results. Show works

    1. (15 pts) (See textbook page 82 for definitions of the economic functions used in this problem.)
      The cost, in dollars, for a company to produce x widgets is given by C(x) = 5250 + 7.00x for
      x ³ 0, and the price-demand function, in dollars per widget, is p(x) = 45 – 0.02x for 0 £ x £ 2250.

      (a) Find and interpret C(300).

      (b) Find and interpret C ̅(300). (Note that C ̅(x) is the average cost function.)

      (c) Find and simplify the expression for the revenue function R(x). (work optional)

      (d) Find and simplify the expression for the profit function P(x). (work optional) Note that p(x) and P(x) are different functions.

      (e) Find and interpret P(300), where P(x) is the profit function in part (d).</x≤7,[email protected]+0.050(x-7,500)&if>

 

Subject Statistics Pages 12 Style APA

Answer

Q1. Which of the graphs describe y as a function of x.

From the given graphs, those that describe y as a function of x i.e. graphs which are functions are graphs B and C. this is because a graph is said to be a function if a vertical line drawn parallel to the y axis intersects the curve only ones. Thus, graphs A and D do not describe functions because a vertical drawn on the graphs would intersect the curves, four times and two times, respectively suggesting that their x-values have more than one output. As a rule of thumb, functions have one output value (y value) for every one input (x value).

Q2. Consider the points (-5, -6) and (-1, 10).

  1. Midpoint of the line with the given end points

Midpoint of a straight line with given endpoints is expressed as;

M = ,        Where M is the midpoint, and x and y the given points.

Thus, with (-5, -6) and (-1, 10) as the given endpoints midpoint of the line can be   computed as;

 M =           = (-6/2, 4/2)

Hence, midpoint of the line is (-3, 2).

  1. Length of the radius of circle using (-3, 2) as the center and (-5, -6) and (-1, 10) as points on the circle.

Distance between (-3, 2) and either of the points (-5, -6) and (-1, 10) i.e. radius of the circle can be obtained the Pythagorean theorem. Because (-2, 2) is midway between the endpoints, point (-1, 10) can be used as the other point between which to find distance.

Thus, we have;

 

 

 

 

 

 

 

 

From the above figure, distance of the radius of the circle formed by the two points can be computed as;

Distance =            =

            = √ (4+64) = √ 68 = 8.25 Units.

Q3. From the following graph of y = f(x), state;

  1. X-intercept

Being a cubic function, the above graph has three x-intercepts i.e. points on the graph where y-value is zero. The intercepts are;

(-1, 0), (1, 0) and (-5, 0).

  1. Y-intercept

y-intercept of the above function i.e. point of the graph where the values of x is zero is;

(0, -3)

  1. Domain of the graph

Domain of the above function is (-1, 6]. This is because it extends horizontally from -1 to 6. 

  1. Range

Range of the above graph is [-3, 2) because it has a vertical extent of 2 to -3.

Q4. Given  ;

  1. Find f (-2)

Substituting x with -2 in the function;

We have f (-2) =  = -3/100

            f (-2) =

  1. State domain of ;

Setting the denominator to 0 and solving for x, we have;

(x – 8)2 = 0; which expands to

x2 – 16x + 64 = 0.

Solving using quadratic formula:  ;

The roots for the equation become;  = (18 ± 0)/2 = 8

Thus, the domain of ;  = all real numbers of x except 8

In notational form, the domain is (∞, 8) U (8, ∞).

  1. Evaluate f (a + 7)

Substituting x in the function with (a + 7), we get;

f (a + 7) =  =  =

 

Q5. Domain of the quotient function given  and

            Replacing x in the function g(x) with -7;

            We have g(x) = g(-7) = |-7 + 7| = |0| = 0

            Hence, domain of the quotient function is (-∞, -7) U (-7, ∞).

            This means the domain of the quotient are all real values of x except -7.

Question 6

  1. Income of $6,300 lies at 2500<x≤7500

Therefore, T(x)= T($6,300)=70+0.035(x-2,500)=70+0.035(6,300-2,500)=$203

  1. Income of $63,000 lies at x>7,500

Therefore, T(x)=T(63,000)=245+0.050(63,000-7,500)= $3,020

 Q7. Let

  1. X-intercepts of the graph

Given that the value of y is zero at the x-intercept i.e. the line y = 0;

We have, 2 – 2x2 = 0; simplifying and solving the equation leads to;

2x2 = 2; =which translates to x2 = 1; and hence, x = ±1.

Thus, x-intercepts of the graph are (1, 0) and (-1, 0).

  1. Y-intercept of the graph.

Given that x value is always zero at the y-intercept; i.e. the line x = 0, we equate 0 in the function;

Hence, we have y = 2 – (2*0)2 = 2;

This means the y-intercept of the graph is (0, 2)

  1. Table of sample points on the graph of the equation

x

y=2-2x2

(x, y)

-4

-30

(-4, 30)

-2

-6

(-2, -6)

0

2

(0, 2)

1

0

(1, 0)

2

-6

(2, -6)

4

-30

(4, -30)

6

-70

(6, -70)

Figure 1: Graph of the equation y=2-2x2 drawn in desmos graphing calculator.

 

  1. Symmetry of the graph: the graph is symmetrical with respect to the y axis at points (-1, 0) and (1, 0); (-6, 0) and (6, 0) among other points.
  2. The graph is not symmetrical around the x-axis.
  3. The graph is symmetrical with respect to the origin as it passes through points (-1, 0) and (1, 0).

Q8. Let and

  1. Evaluate g – f for x = -3

f(x) = 4x2 + 2x – 8;            therefore f(-3) = 4(-3)2 + 2*(-3) – 8 = 22

g(x) = 1 – 2x;                      therefore g(-3) = 1- (2*-3) = 7

thus, (g – f)(-3) = 7 – 22 = -15.

  1. Evaluate fg for x = -3 i.e find fg(-3)

f(x) = 4x2 + 2x – 8;            This implies f(-3) = 4(-3)2 + 2*(-3) – 8 = 22

                g(x) = 1 – 2x;                      This implies g(-3) = 1- (2*-3) = 7

                Hence fg(-3) = 22*7 = 154

  1. Find the difference function (f – g) (x)

(f – g) (x) = (4x2 + 2x – 8) – (1 – 2x)

                = 4x2 + 2x +2x – 8 – 1

Thus, (f – g) (x) = 4x2 + 4x – 9

Question 9

  1. Cost C(x) =5,250+7.00x for x≥0

When x=300, C(x)=C(300)=5250+7.00(300)=$7,350

This implies that the total cost of producing 300 widgets is $7,350

  1. Average cost ‘C(x)=Total cost/Number of units produced

for X=300, ‘C(x)= ‘C(300)=7,350/300=$245

This means that the average cost of producing each unit of widget is $245. However, the average cost does not represent the actual cost of production for the item.

  1. Revenue Function=price-demand function*total output in units

For x units, R(x)=x*p(x)=x(45-0.02x) for 0≤x≤2,250

R(x)=45x-0.02x2 for 0≤x≤2,250

  1. Profit=Total Revenue-Total Cost

Profit function, P(x)=45x-0.02x2 -5,250-7.00x for 0≤x≤2,250

P(x)=-0.02x2+38x-5,250

  1. For 300 units, P(x)=-(0.02*3002)+(38*300)-5,250=$4,350

This implies that by producing and selling 300 widgets, the company gains a total of $4,350

 

References

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