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null hypothesis, because Ξ± β€ p. 41. Reject the null hypothesis, because Ξ± β₯ p. 42. H0: ΞΌ β₯ 29.0β Ha: ΞΌ < 29.0β 43. t19. Because you do not know the population standard deviation, use the t distribution. The degrees of freedom are 19, because df = n β 1. 44. The test statistic is β4.4721 and the p-value is 0.00013 usin... |
the mean difference in amount spent on textbooks for the two groups Β― 60. H0: X Β― Ha: X 2 This could also be written as > 0 Β― H0: X Β― Ha 61. Using the calculator function 2-SampTTest, reject the null hypothesis. At the five percent significance level, there is sufficient evidence to conclude that the science students ... |
55 β 68 = β13 169 Enrolled 145 Not enrolled 55 Table B13 75. df = n β 1 = 2 β 1 = 1. (O β E)2 z 169 132 169 68 = 1.280 = 2.485 76. Using the calculator function Chi-Square GOF Test (in STAT TESTS), the test statistic is 3.7656 and the p-value is 0.0523. Do not reject the null hypothesis. At the five percent significan... |
The populations do not have the same distribution. 11.6: Test of a Single Variance 88. H0: Ο2 β€ 5 Ha: Ο2 > 5 Practice Test 4 12.1 Linear Equations 1. Which of the following equations is/are linear? A. y = β3x B. y = 0.2 + 0.74x C. y = β9.4 β 2x D. A and B E. A, B, and C 2. To complete a painting job requires four hour... |
ression Equation Use the following information to answer the next four exercises. Height (in inches) and weight (in pounds) in a sample of college freshman males have a linear relationship with the following summary statistics: xΒ― = 68.4 yΒ― =141.6 sx = 4.0 sy = 9.6 r = 0.73 Let Y = weight and X = height, and write the ... |
serving? 12.8: Outliers 24. In the context of regression analysis, what is the definition of an outlier, and what is a rule of thumb to evaluate if a given value in a data set is an outlier? 25. In the context of regression analysis, what is the definition of an influential point, and how does an influential point dif... |
Distribution Use the following information to answer the next seven exercises. You are conducting a study of three types of feed supplements for cattle to test their effectiveness in producing weight gain among calves whose feed includes one of the supplements. You have four groups of 30 calves (one is a control group... |
B 869 Practice Test 4 Solutions 12.1 Linear Equations 1. e. A, B, and C. All three are linear equations of the form y = mx + b. 2. Let y = the total number of hours required, and x the square footage, measured in units of 1,000. The equation is y = x + 4 3. Let y = the total payment, and x the number of students in a ... |
15. y^ = 21.76 + 1.75(68) = 140.76 12.5: Correlation Coefficient and Coefficient of Determination 16. The coefficient of determination is the square of the correlation, or r2. For this data, r2 = (β0.56)2 = 0.3136 β 0.31 or 31 percent. This means that 31 percent of the variation in fuel efficiency can be explained by ... |
+ 16(5) = 105 23. Because the intercept appears in both predicted values, you can ignore it in calculating a predicted difference score. The difference in grams of fiber per serving is 6 β 3 = 3, and the predicted difference in grams of potassium per serving is (16)(3) = 48. 12.8: Outliers 24. An outlier is an observe... |
the outcome is a numerical variable (test score), and you were told the samples were independent and randomly selected, so those requirements are met. However, each sample has a different standard deviation, and this suggests that the populations from which they were drawn also have different standard deviations, whic... |
13.3: Facts About the F Distribution 40. All but choice c, β3.61. F Statistics are always greater than or equal to 0. 41. As the degrees of freedom increase in an F distribution, the distribution becomes more nearly normal. Histogram F2 is closer to a normal distribution than histogram F1, so the sample displayed in h... |
20 10 Table B15 22 30 28 8 4. Find the probability that a randomly chosen student is male or has a bedroom painted with light colors. A. B. C. D. 10 118 68 118 48 118 10 48 5. Find the probability that a randomly chosen student is male given the studentβs bedroom is painted with dark colors. A. B. C. 30 118 30 48 22 1... |
students who enroll in math classes. In a sample of 1,795 students enrolled in Math 1A (1st quarter calculus), 1,428 passed the course. In a sample of 856 students enrolled in Math 1B (2nd quarter calculus), 662 passed. In general, are the pass rates of Math 1A and Math 1B statistically the same? Let A = the subscript... |
are from the Gorsuch Ltd. Winter catalog: $212, $292, $278, $199, $280, $236. Assume the underlying sweater price population is approximately normal. The null hypothesis is that the mean price of adult ski sweaters from Gorsuch Ltd. is at least $275. 15. Which of the following is the correct distribution to use for th... |
week? A. 49 B. 25 C. 30 D. 13 22. What is the 60th percentile for this data? A. 2 B. 3 C. 4 D. 5 Use the following information to answer the next two exercises. The following data are the results of a random survey of 110 reservists called to active duty to increase security at California airports. Number of Dependent... |
.5927, 0.8057) (0.6312, 0.7672) 29. Winning times in 26 mile marathons run by world class runners average 145 minutes with a standard deviation of 14 minutes. A sample of the last 10 marathon winning times is collected. Let x = mean winning times for 10 marathons. The distribution for x is A. N β β145, 14 10 β β B. N(1... |
. One question asked which major within the business program the student had chosen. Here are the data from the students who responded. Female Male Accounting Administration Economics Finance 68 91 5 61 56 40 6 59 Table B21 Does the data suggest that there is a relationship between the gender of students and their choi... |
80 14. C Iris 15. C Student's t 16. B is left-tailed. 17. C cluster sampling 18. B median 19. A the probability that an outcome of the data will happen purely by chance when the null hypothesis is true. Appendix B 20. D stratified 21. B 25 22. C 4 23. A (1.85, 2.32) 24. C Both above are correct. 25. C 5.8 26. C 0.6321 ... |
B. Drought and water rationing are mutually exclusive events. C. None of the above. 882 Appendix B Use the following information to answer the next two exercises. Suppose that a survey yielded the following data: gender apple pumpkin pecan female male 40 20 10 30 Table B23 Favorite Pie 30 10 6. Suppose that one indivi... |
answer the next three exercises. The amount of money a customer spends in one trip to the supermarket is known to have an exponential distribution. Suppose the average amount of money a customer spends in one trip to the supermarket is $72. 13. What is the probability that one customer spends less than $72 in one trip... |
5 21. Find the probability that the average wait time for ten students is at most 5.5 minutes. A. 0.6301 B. 0.8541 C. 0.3694 D. 0.1459 22. A sample of 80 software engineers in Silicon Valley is taken, and it is found that 20 percent of them earn approximately $50,000 per year. A point estimate for the true proportion o... |
C. There is sufficient evidence to conclude that the proportion for males is higher than the proportion for females. D. There is not enough information to make a conclusion. 27. From past experience, a statistics teacher has found that the average score on a midterm is 81, with a standard deviation of 5.2. This term, ... |
0: Ο2 = 1.6 vs. Ha: Ο2 β 1.6. Which graph best describes the results of the test? Figure B13 Sixty-four backpackers were asked the number of days since their latest backpacking trip. The number of days is given in Table B26. # of days 1 2 3 4 5 6 7 8 Frequency 5 9 6 12 7 10 5 10 Table B26 33. Conduct an appropriate tes... |
β 0.0094 31. D We should not be estimating here. 32. A 33. A The p-value is > 0.10. There is insufficient information to conclude that the distribution is not uniform. 34. C The test is to determine if the different groups have the same means. This OpenStax book is available for free at http://cnx.org/content/col30309... |
132 133 127 138 126 137 131 131 Table C2 Practice Lap Times (in seconds) Stock Prices The following table lists initial public offering (IPO) stock prices for all 1999 stocks that at least doubled in value during the first day of trading. $17.00 $23.00 $14.00 $16.00 $12.00 $26.00 $20.00 $22.00 $14.00 $15.00 $22.00 $18... |
12.00 $8.00 $23.00 $12.00 $18.00 $20.00 $21.00 $34.00 $16.00 $26.00 $14.00 Table C3 IPO Offer Prices References Data compiled by Jay R. Ritter of University of Florida using data from Securities Data Co. and Bloomberg. 892 Appendix C This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Append... |
concentrated together or spread apart? Explain how you determined this. ____ Looking at both the histogram and the box plot, discuss the distribution of your data. Assignment Checklist You need to turn in the following typed and stapled packet, with pages in the following order: ____ Cover sheet: name, class time, and... |
and highest values as a and b. ____ Normal: X ~ N ____________ Use xΓΒ― to estimate for ΓΒΌ and s to estimate for ΓΖ. ____ Must your data fit one of the above distributions? Explain why or why not. ____ Could the data fit two or three of the previous distributions (at the same time)? Explain. ____ Calculate the value k(... |
Β―. X ΓΒ― ~ ______________ ΓΒ― (an X ΓΒ― value) that is 1.75 standard deviations above the sample mean. k ΓΒ― = _____ (rounded to Calculate the value k two decimal places) Determine the relative frequencies (RF) rounded to four decimal places. a. RF( X ΓΒ― < k ΓΒ― ) = _______ b. RF( X ΓΒ― > k ΓΒ― ) = _______ c. RF( X ΓΒ― = k ΓΒ― ... |
hesis Testing-Article Student Learning Objectives β’ The student will identify a hypothesis testing problem in print. β’ The student will conduct a survey to verify or dispute the results of the hypothesis test. β’ The student will summarize the article, analysis, and conclusions in a report. Instructions As you complete ... |
As previously mentioned, convenience sampling is not acceptable. d. Conclusion about the article claim in light of your hypothesis test; this is the conclusion of your hypothesis test, stated in words, in the context of the situation in your project in sentence form, as if you were writing this conclusion for a non-st... |
, stratified, systematic, or simple random sampling (using a random number generator) sampling. Convenience sampling is NOT acceptable. ____Conduct your survey. Your number of pairs must be at least 30. ____Print out a copy of your data. Analysis ____On a separate sheet of paper construct a scatter plot of the data. La... |
by age as 20-26,27-33,34-40,41-47,48-54,55-61... Instead, maybe use age groups 19.5-24.5, 24.5-29.5,... or 19.5-29.5, 29.5-39.5, 39.5-49.5,... _____In complete sentences, describe the shape of your histogram. _____Are there any potential outliers? Which values are they? Show your work and calculations as to how you us... |
content/col30309/1.8 Appendix E 899 APPENDIX E: SOLUTION SHEETS Hypothesis Testing With One Sample Class Time: __________________________ Name: _____________________________________ a. H0: _______ b. Ha: _______ c. In words, clearly state what your random variable X Β― or Pβ² represents. d. State the distribution to use ... |
__ Name: ____________________________________ a. H0: _______ b. Ha: _______ c. What are the degrees of freedom? d. State the distribution to use for the test. e. What is the test statistic? f. What is the p-value? In one to two complete sentences, explain what the p-value means for this problem. g. Use the previous inf... |
the same as 4. X is not 4. X is not equal to 4. X is not the same as 4. X is different than 4. Table F1 Formulas Formula 1: Factorial 904 Appendix F n! = n(n β 1)(n β 2)...(1) 0! = 1 Formula 2: Combinations β β = n β β r n! (n β r)!r! Formula 3: Binomial Distribution X ~ B(n, p) P(X = x) = n β β x β β p x qn β x, for x... |
n = degrees of freedom Formula 12: Chi-Square Distribution 2 X ~ Ξ§d f f (x) = x 2 n β 2 β = positive integer and degrees of freedom Formula 13: F Distribution X ~ Fd f (n), d f (d) d f (n) = degrees of freedom for the numerator d f (d) = degrees of freedom for the denominator f (x) = Ξ(u + v 2 ) 2)Ξ( v Ξ(u 2) (u 2 u 2... |
complement of A same green on first pick same prob. of green on first pick same Discrete Random Variables PDF prob. distribution function same Discrete Random Variables X X the random variable X Discrete Random Variables X ~ the distribution of X Discrete Random Variables B binomial distribution Discrete Random Variab... |
; p-hat sample proportion of success q-prime; q-hat sample proportion of failure same same same same same same same same same same same same same 908 Appendix F Chapter (1st used) Symbol Spoken Meaning Hypothesis Testing Hypothesis Testing Hypothesis Testing Hypothesis Testing Hypothesis Testing Hypothesis Testing Hypo... |
notation Numbers in scientific notation are expressed on the TI-83, 83+, 84, and 84+ using E notation, such that... β’ 4.321 E 4 = 4.321Γ104 β’ 4.321 E β4 = 4.321Γ10β4 To transfer programs or equations from one calculator to another Both calculators: Insert your respective end of the link cable cable and press, then [LI... |
the same manner until all data values are entered. 11. Access statistics mode. 12. Navigate to <CALC>. 13. Access <1:1-var Stats>. 14. Indicate that the data is in [L1]..., [L1],, 15....and indicate that the frequencies are in [L2]., [L2],. 16. The statistics should be displayed. You may arrow down to get remaining st... |
β¦ Xscl = 1 (width of bars) β¦ Ymin = 0 β¦ Ymax = 10 β¦ Yscl = 1 (spacing of tick marks on y-axis) β¦ Xres = 1 3. Access graphing mode to see the histogram. To draw box plots 1. Access graphing mode., [STAT PLOT]. 2. Select <1:Plot 1> to access the first graph. 3. Use the arrows to select <ON> and turn on Plot 1. 4. Use th... |
Navigate to <Float> and then to the right until you reach <3>. β¦ All numbers will be rounded to three decimal places until changed.. 4. Enter statistics mode and clear lists [L1] and [L2], as described previously. 916 Appendix G,. 5. Enter editing mode to insert values for x and y.,. 6. Enter each value. Press to cont... |
Ymax = 60 β¦ Yscl = 10 (spacing of tick marks on y-axis) β¦ Xres = 1 14. Be sure to deselect or clear all equations before graphing, using the instructions above. 15. Press the graph button to see the scatter plot. To see the regression graph 1. Access the equation menu. The regression equation will be put into Y1. 918 ... |
One way is to substitute values for "x" in the equation. Another way on the graph of the regression line. is to use the TI-83, 83+, 84, 84+ instructions for distributions and tests Distributions Access DISTR for Distributions. For technical assistance, visit the Texas Instruments website at http://www.ti.com (http://w... |
num,dfdenom) yields the probability density function value, only useful to plot the F curve, in which case "x" is the variable β’ Fcdf(left bound,right bound,dfnum,dfdenom) corresponds to P(left bound < F < right bound) Tests and Confidence Intervals Access STAT and TESTS. For the confidence intervals and hypothesis tes... |
the back button on your browser to return here. Tables (NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, January 3, 2009) β’ Student t table (http://www.itl.nist.gov/div898/handbook/eda/section3/eda3672.htm) β’ Normal table (http://www.itl.nist.gov/div898/handbook/eda/section3/e... |
), 490, 552 confidence intervals, 477 Confidence intervals, 523 confidence level, 461, 477 confidence level (CL), 490 contingency table, 203, 225, 649, 667 continuity correction factor, 430 continuous, 10 continuous random variable, 47, 340 control group, 38, 47 convenience sampling, 47 critical value, 388 cumulative d... |
225 N nominal scale, 29 nonsampling error, 47 normal distribution, 399, 439, 472, 490, 529, 552 normally distributed, 415, 419, 529 924 Index the AND event, 225 the complement event, 225 the conditional probability of one event GIVEN another event, 225 the law of large numbers, 294 the OR event, 225 the OR of two even... |
, 415 sample size, 415 sample space, 198, 211, 225 samples, 27 sampling, 7 sampling bias, 48 sampling distribution, 109, 439 sampling error, 48 sampling variability of a statistic, 119 sampling with replacement, 48, 225 sampling without replacement, 48, 225 simple random sample, 529 simple random sampling, 48 skewed, 1... |
(x1, y0) With a little more imagination, we can envision a right triangle whose hypotenuse has length d as drawn above on the right. From the latter ο¬gure, we see that the lengths of the legs of the triangle are |x1 β x0| and |y1 β y0| so the Pythagorean Theorem gives us |x1 β x0|2 + |y1 β y0|2 = d2 (x1 β x0)2 + (y1 β... |
x0)2 + (y1 β y0)2 (1 β 3)2 + (y β 2)2 4 + (y β 2)2 4 + (y β 2)22 4 = 4 = 42 = 16 = 4 + (y β 2)2 12 = (y β 2)2 β (y β 2)2 = 12 2 12 β 3 β y = 2 Β± 2 3 squaring both sides extracting the square root We obtain two answers: (1, 2 + 2 why there are two answers. β 3) and (1, 2 β 2 β 3). The reader is encouraged to think abou... |
. If a = b, prove that the line y = x equally divides the line segment with endpoints (a, b) and (b, a). Solution. To prove the claim, we use Equation 1.2 to ο¬nd the midpoint Since the x and y coordinates of this point are the same, we ο¬nd that the midpoint lies on the line y = x, as required. 14 Relations and Function... |
8), E(β5.5, 0), F (β8, 4), G(9.2, β7.8) and H(7, 5) in the Cartesian Coordinate Plane given below9 β8 β7 β6 β5 β4 β3 β2 β1 β2 β3 β4 β5 β6 β7 β8 β9 21. For each point given in Exercise 20 above Identify the quadrant or axis in/on which the point lies. Find the point symmetric to the given point about the x-axis. Find t... |
The points are arranged horizontally. (Hint: Use P (x0, b) and Q(x1, b).) (c) The points are actually the same point. (You shouldnβt need a hint for this one.) 36. Verify the Midpoint Formula by showing the distance between P (x1, y1) and M and the distance between M and Q(x2, y2) are both half of the distance between... |
1 0 2 β5 β3 5 4 β3 2. (β1, 5] β© [0, 8) = [0, 5] 3. (β1, 1) βͺ [0, 6] = (β1, 6] 4. (ββ, 4] β© (0, β) = (0, 4] 5. (ββ, 0) β© [1, 5] = β
6. (ββ, 0) βͺ [1, 5] = (ββ, 0) βͺ [1, 5] 7. (ββ, 5] β© [5, 8) = {5} 8. (ββ, 5) βͺ (5, β) 9. (ββ, β1) βͺ (β1, β) 10. (ββ, β3) βͺ (β3, 4) βͺ (4, β) 11. (ββ, 0) βͺ (0, 2) βͺ (2, β) 12. (ββ, β2) βͺ (β2, ... |
4 β5 β6 β7 β8 β9 A(β3, β7) G(9.2, β7.8) 1.1 Sets of Real Numbers and the Cartesian Coordinate Plane 19 21. (a) The point A(β3, β7) is (b) The point B(1.3, β2) is in Quadrant III symmetric about x-axis with (β3, 7) symmetric about y-axis with (3, β7) symmetric about origin with (3, 7) in Quadrant IV symmetric about x-ax... |
d = 5, M = β1, 7 2 26, M = 1, 3 2 24. d = β β β 26. d = 28. d = 30. (3 + 74, M = 13 10, β 13 β 10 83, M = 4 β 2 5, 5 β β 7, β1), (3 β β 32. (β1 + 3, 0), (β1 β 7, β1) β 3, 0) 34. (β3, β4), 5 miles, (4, β4) 3 29. d = β 23. d = 4 β 25. d = 27. d = 3 37 10, M = (1, β4, β x2 + y2, M = x 5 31. (0, 3) β 33 37. (a) The distan... |
points. Doing so produces the graph of R. y 4 3 2 1 (β2, 1) (4, 3) β4 β3 β2 β1 1 2 3 4 x β1 β2 β3 β4 (0, β3) The graph of R. In the following example, we graph a variety of relations. Example 1.2.1. Graph the following relations. 1. A = {(0, 0), (β3, 1), (4, 2), (β3, 2)} 2. HLS1 = {(x, 3) | β 2 β€ x β€ 4} 3. HLS2 = {(x,... |
Ο, 3), and so on. 6, 34 β3 β2 β1 1 2 3 4 x β4 β3 β2 β1 1 2 3 4 x The graph of A The graph of HLS1 3. HLS2 is hauntingly similar to HLS1. In fact, the only diο¬erence between the two is that instead of ββ2 β€ x β€ 4β we have ββ2 β€ x < 4β. This means that we still get a horizontal line segment which includes (β2, 3) and ex... |
{(x, y) | y = β2} is similar to the relation V above in that only one of the coordinates, in this case the y-coordinate, is speciο¬ed, leaving x to be βfreeβ. Plotting some representative points gives us the horizontal line y = β2. y 4 3 2 1 β1 β2 β3 β4 1 2 3 4 x y β4 β3 β2 β1 1 2 3 4 x β1 β2 β3 β4 The graph of H The g... |
learned the following. Equations of Vertical and Horizontal Lines The graph of the equation x = a is a vertical line through (a, 0). The graph of the equation y = b is a horizontal line through (0, b). Given that the very simple equations x = a and y = b produced lines, itβs natural to wonder what shapes other equatio... |
in the following example. Example 1.2.3. Graph x2 + y3 = 1. Solution. To eο¬ciently generate points on the graph of this equation, we ο¬rst solve for y x2 + y3 = 1 y3 = 1 β x2 y3 = 3β y = 3β We now substitute a value in for x, determine the corresponding value y, and plot the resulting point (x, y). For example, substit... |
come in two distinct varieties: x-intercepts and y-intercepts. They are deο¬ned below. Deο¬nition 1.5. Suppose the graph of an equation is given. A point on a graph which is also on the x-axis is called an x-intercept of the graph. A point on a graph which is also on the y-axis is called an y-intercept of the graph. In ... |
, y) satisο¬es the equation (since it leads to a true result) and hence is on the graph. In this way, we can check whether the graph of a given equation possesses any of the symmetries discussed in Section 1.1. We summarize the procedure in the following result. Testing the Graph of an Equation for Symmetry To test the ... |
that the graph is symmetric about the y-axis or the origin. If the graph possessed either of these symmetries, then the fact that (1, 0) is on the graph would mean (β1, 0) would have to be on the graph. (Why?) Since (β1, 0) would be another x-intercept (and weβve found all of these), the graph canβt have y-axis or ori... |
to be equivalent to our original equation. However, to actually prove that the graph is not symmetric about the y-axis, we need to ο¬nd a point (x, y) on the graph whose reο¬ection (βx, y) is not. Our x-intercept (1, 0) ο¬ts this bill nicely, since if we substitute (β1, 0) into the equation we get (x β 2)2 + y2 (β1 β 2)2... |
{(x, y) | x > 0, y < 4} 20. {(x, y } In Exercises 21 - 30, describe the given relation using either the roster or set-builder method. 21. y 4 3 2 1 β4 β3 β2 β1 β1 1 x Relation A 22. y 3 2 1 β4 β3 β2 β1 1 2 3 4 x Relation B 30 23. y 5 4 3 2 1 β1 β2 β3 1 2 3 x Relations and Functions 24. y 3 2 1 β3 β2 β1 β1 x β2 β3 β4 R... |
), (8, 3), (16, 4), (32, 5),...} 40. {..., (β3, 9), (β2, 4), (β1, 1), (0, 0), (1, 1), (2, 4), (3, 9),...} For each equation given in Exercises 41 - 52: Find the x- and y-intercept(s) of the graph, if any exist. Follow the procedure in Example 1.2.3 to create a table of sample points on the graph of the equation. Plot t... |
)2 = x3 + y3 Crooked egg 57. With the help of your classmates, ο¬nd examples of equations whose graphs possess symmetry about the x-axis only symmetry about the y-axis only symmetry about the origin only symmetry about the x-axis, y-axis, and origin Can you ο¬nd an example of an equation whose graph possesses exactly two... |
= {(x, y) | x > β2} 28. H = {(x, y) | β 3 < x β€ 2} 29. I = {(x, y) | x β₯ 0,y β₯ 0} 30. J = {(x, y) | β 4 < x < 5, β3 < y < 2} 31. 33. 35. y 3 2 1 β3 β2 β1 β1 x β2 β3 The line x = β2 y 3 2 1 β3 β2 β1 1 2 3 x The line y = 3 y 3 2 1 32. y 3 2 1 β1 β2 β3 1 2 3 x The line x = 3 34. y β3 β2 β1 β1 1 2 3 x β2 β3 The line y = β... |
8 3 β3β2β1 β2 β3 β4 β5 β6 β7 β8 β9 The graph is not symmetric about the x-axis (e.g. (β3, 7) is on the graph but (β3, β7) is not) The graph is not symmetric about the y-axis (e.g. (β3, 7) is on the graph but (3, 7) is not) The graph is not symmetric about the origin (e.g. (β3, 7) is on the graph but (3, β7) is not) 38 ... |
) is on the graph but (4, β4) is not) The graph is symmetric about the origin 1.2 Relations 45. y = β x β 2 x-intercept: (2, 0) The graph has no y-intercepts β 46-intercept: (β3, 0) y-intercept: (0, 2) 39 y (x, y) 0 (2, 0) 1 (3, 1) (6, 2) 2 3 (11, 3) x 2 3 6 11 10 11 x The graph is not symmetric about the x-axis (e.g. ... |
β4 2 β1 2 3 y 3 2 1 β2β1 β1 1 2 3 x β2 β3 β4 β5 β6 β7 β8 β9 β10 β11 β12 β13 x-intercepts: 10 3, 0 y-intercept: (0, β5) x y β2 β8 β1 β 13 2 0 β5 1 β 7 2 2 β2 (x, y) (β2, β8) β1, β 13 2 (0, β5) 1, β 7 2 (2, β2) y 2 1 β3β2β1 β1 1 2 3 4 x β2 β3 β4 β5 β6 β7 β8 β9 The graph is not symmetric about the x-axis (e.g. (3, 2) is ... |
. x-intercepts: (β1, 0), (1, 0) The graph has no y-intercepts x β3 Β± β2 Β± βx, y) β β 8 (β3, Β± 3 (β2, Β± 8) 3) (β1, 0) (1, 0) β β (2, Β± (3, Β± 3) 8) 3 8 β7β6β5β4β3β2β1 β1 1 2 3 x β2 β3 β4 β5 The graph is symmetric about the x-axis The graph is not symmetric about the y-axis (e.g. (β6, 0) is on the graph but (6, 0) is not)... |
is symmetric about the y-axis The graph is symmetric about the origin The graph is not symmetric about the x-axis (e.g. (1, β4) is on the graph but (1, 4) is not) The graph is not symmetric about the y-axis (e.g. (1, β4) is on the graph but (β1, β4) is not) The graph is symmetric about the origin 1.3 Introduction to F... |
graphically as the points (1, 3) and (1, 4) lying on the same vertical line, x = 1. If we turn our attention to the graph of R2, we see that no two points of the relation lie on the same vertical line. We can generalize this idea as follows Theorem 1.1. The Vertical Line Test: A set of points in the plane represents y... |
is a point on the curve with x-coordinate 1 just below (1, 2), which means that both (1, 2) and this point on the curve lie on the vertical line x = 1. (See the picture below and the left.) Hence, the graph of S1 fails the Vertical Line Test, so y is not a function of x here. However, in S2 notice that the point with ... |
we need to pay attention to two 46 Relations and Functions subtle notations on the graph: the arrowhead on the lower left corner of the graph indicates that the graph continues to curve downwards to the left forever more; and the open circle at (1, 3) indicates that the point (1, 3) isnβt on the graph, but all points ... |
1) If we substitute x = 0 into our equation for y, we get y = Β± and (0, β1) are on the graph of this equation. Hence, this equation does not represent y as a function of x. β x2 + y3 = 1 3 y3 = 1 β x2 y3 = 3β y = 3β 1 β x2 1 β x2 For every choice of x, the equation y = 3β equation describes y as a function of x. 1 β x... |
), (β1, 1), (0, 0), (1, 1), (2, 4), (3, 9)} 2. {(β3, 0), (1, 6), (2, β3), (4, 2), (β5, 6), (4, β9), (6, 2)} 3. {(β3, 0), (β7, 6), (5, 5), (6, 4), (4, 9), (3, 0)} 4. {(1, 2), (4, 4), (9, 6), (16, 8), (25, 10), (36, 12),...} 5. {(x, y) | x is an odd integer, and y is an even integer} 6. {(x, 1) | x is an irrational numbe... |
4β3β2β1 β1 1 2 3 4 5 x β2 β3 β4 β5 1.3 Introduction to Functions 51 23. 25. 27. 29. y 5 4 3 2 1 β5β4β3β2β1 β1 1 2 3 4 5 x β2 β3 β4 β5 y 4 3 2 1 24. y 5 4 3 2 1 β1 β1 β2 β3 β4 β5 262 β1 1 2 x β2 β1 1 2 x y 4 3 2 1 28. y 4 3 2 1 β2 β1 1 2 x β2 β1 1 2 x y 2 1 30. y 2 1 β2 β1 1 2 x β3 β2 β1 1 2 3 x β1 β2 β1 β2 52 31. y 2 1... |
Test 1.1. Discuss with your classmates how you might ο¬nd such points for the relations given in Exercises 50 - 53. 50. x3 + y3 β 3xy = 0 52. y2 = x3 + 3x2 51. x4 = x2 + y2 53. (x2 + y2)2 = x3 + y3 1.3 Introduction to Functions 53 1.3.2 Answers 1. Function 2. Not a function domain = {β3, β2, β1, 0, 1, 2,3} range = {0, ... |
[β3, β) 23. Function domain = [β5, 4) range = [β4, 4) 25. Function domain = (ββ, β) range = (ββ, 4] 27. Function domain = [β2, β) range = (ββ, 3] 29. Function domain = (ββ, 0] βͺ (1, β) range = (ββ, 1] βͺ {2} 31. Not a function 22. Not a function 24. Function domain = [0, 3) βͺ (3, 6] range = (β4, β1] βͺ [0, 4] 26. Functi... |
Domain (Inputs) y = f (x) Range (Outputs) The value of y is completely dependent on the choice of x. For this reason, x is often called the independent variable, or argument of f, whereas y is often called the dependent variable. As we shall see, the process of a function f is usually described using an algebraic form... |
key skill for success in this and many other Math courses. Example 1.4.2. Let f (x) = βx2 + 3x + 4 1. Find and simplify the following. (a) f (β1), f (0), f (2) (b) f (2x), 2f (x) (c) f (x + 2), f (x) + 2, f (x) + f (2) 2. Solve f (x) = 4. Solution. 1. (a) To ο¬nd f (β1), we replace every occurrence of x in the expressi... |
6 = βx2 + 3x + 10 2. Since f (x) = βx2 + 3x + 4, the equation f (x) = 4 is equivalent to βx2 + 3x + 4 = 4. Solving we get βx2 + 3x = 0, or x(βx + 3) = 0. We get x = 0 or x = 3, and we can verify these answers by checking that f (0) = 4 and f (3) = 4. A few notes about Example 1.4.2 are in order. First note the diο¬eren... |
3, β). When a formula for a function is given, we assume that the function is valid for all real numbers which make arithmetic sense when substituted into the formula. This set of numbers is often called the implied domain2 of the function. At this stage, there are only two mathematical sins we need to avoid: division ... |
. 1.4 Function Notation 59 1 β 4x x β 3 = 0 1 = (1)(x β 3) = 4x x β 3 4x x β 3 x β 3 = 4x β3 = 3x β1 = x (x β 3) clear denominators So we get two real numbers which make denominators 0, namely x = β1 and x = 3. Our domain is all real numbers except β1 and 3: (ββ, β1) βͺ (β1, 3) βͺ (3, β). 4. In ο¬nding the domain of F, we... |
β3, 33) βͺ (33, β). t + 3 = 0, or. Itβs tempting to simplify I(x) = 3x2 x = 3x, and, since there are no longer any denominators, claim that there are no longer any restrictions. However, in simplifying I(x), we are assuming x = 0, since 0 0 is undeο¬ned.6 Proceeding as before, we ο¬nd the domain of I to be all real number... |
get c = 1.5g In order for the units to be correct in the formula, g must be measured in pounds of grapes in which case the computed value of c is measured in dollars. Since weβre interested in ο¬nding the cost c given an amount g, we think of g as the independent variable and c as the dependent variable. Using the lang... |
? 1.4 Function Notation 61 these limits determine the applied domain of g. Typically, an applied domain is stated explicitly. In this case, it would be common to see something like c(g) = 1.5g, 0 β€ g β€ 100, meaning the number of pounds of grapes purchased is limited from 0 up to 100. The upper bound here, 100 may repre... |
h(60) = 0. This means that 60 seconds after lift-oο¬, the rocket is 0 feet above the ground; in other words, a minute after lift-oο¬, the rocket has already returned to Earth. 2. Since the function h is deο¬ned in pieces, we need to solve h(t) = 375 in pieces. For 0 β€ t β€ 20, h(t) = β5t2 + 100t, so for these values of t,... |
to model. In that respect, mathematical modeling cannot be a topic in a book, but rather, must be a theme of the book. For now, we have you explore some very basic models in the Exercises because you need to crawl to walk to run. As we learn more about functions, weβll help you build your own models and get you on you... |
real number x and performs the following three steps in the order given: (1) take the square root; (2) make the quantity the denominator of a fraction with numerator 4; (3) subtract 13. 10. f is a function that takes a real number x and performs the following three steps in the order given: (1) make the quantity the d... |
x 33. f (x) = 3 4 β x ο£±  35. Let f (x) =  (a) f (β4) (d) f (3.001) 34. f (x) = 3x2 β 12x 4 β x2 β x + 5 9 β x2 βx + 5 x β€ β3 if if β3 < x β€ 3 x > 3 if Compute the following function values. (b) f (β3) (e) f (β3.001) (c) f (3) (f) f (2) 1.4 Function Notation 65 36. Let f (x) = ο£±   β x2 1 β x2 x x β€ β1 if if β1 <... |
50. f (x) = β 6 6x β 2 6 β 6x β 2 52. f (x) = 54. f (x) = 56. Q(r) = 58. A(x) = 60. g(v) = 62. u(w) = 4 β β 3 6x β 2 x2 + 36 β r r β 8 β 1 4 β 1 v2 66 Relations and Functions 63. The area A enclosed by a square, in square inches, is a function of the length of one of its sides x, when measured in inches. This relation... |
68. The temperature T in degrees Fahrenheit t hours after 6 AM is given by T (t) = β 1 2 t2 + 8t + 3 for 0 β€ t β€ 12. Find and interpret T (0), T (6) and T (12). 69. The function C(x) = x2 β 10x + 27 models the cost, in hundreds of dollars, to produce x thousand pens. Find and interpret C(0), C(2) and C(5). 70. Using d... |
is the cost to ship 10 comic books? (b) What is the signiο¬cance of the formula S(n) = 0 for n β₯ 15? 74. The cost C (in dollars) to talk m minutes a month on a mobile phone plan is modeled by C(m) = 25 25 + 0.1(m β 1000) 0 β€ m β€ 1000 if if m > 1000 (a) How much does it cost to talk 750 minutes per month with this plan?... |
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