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HAPTER 10 | HYPOTHESIS TESTING WITH TWO SAMPLES P′A – P′B follows an approximate normal distribution. Calculate the p-value using the normal distribution: p-value = 0.1404. Estimated proportion for group A: p′ A = Estimated proportion for group B: p′B = x A n A xB nB = 20 200 = 12 200 = 0.1 = 0.06 Graph: Figure 10.7 P′... |
lue. Make a decision: Since α > p-value, reject H0. This means that μd < 0 and there is improvement. Conclusion: At a 5% level of significance, from the sample data, there is sufficient evidence to conclude that the sensory measurements, on average, are lower after hypnotism. Hypnotism appears to be effective in reduci... |
lable for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 10 | HYPOTHESIS TESTING WITH TWO SAMPLES 551 KEY TERMS Degrees of Freedom (df) the number of objects in a sample that are free to vary. Pooled Proportion estimate of the common value of p1 and p2. Stan... |
Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if th... |
ls thought that the mean difference between their jumping and relaxed times would be zero. Test their hypothesis. Relaxed time (seconds) Jumping time (seconds) 26 47 30 22 Table 10.24 21 40 28 21 558 CHAPTER 10 | HYPOTHESIS TESTING WITH TWO SAMPLES Relaxed time (seconds) Jumping time (seconds) 23 45 37 29 Table 10.24 2... |
n the husband’s versus the wife’s satisfaction level is negative (meaning that, within the partnership, the husband is happier than the wife). Wife’s Score Husband’s Score Table 10.26 10.3 Comparing Two Independent Population Proportions DIRECTIONS: For each of the word problems, use a solution sheet to do the hypothes... |
a solution sheet to do the hypothesis test. The solution sheet is found in m47889 (http://cnx.org/content/m47889/latest/) . Please feel free to make copies of the solution sheets. For the online version of the book, it is suggested that you copy the .doc or the .pdf files. NOTE If you are using a Student's t-distributi... |
t of two independent means. b. a test of two proportions. c. a test of a single mean. d. a test of a single proportion. REFERENCES 10.1 Two Population Means with Unknown Standard Deviations Data from Graduating Engineer + Computer Careers. Available online at http://www.graduatingengineer.com Data from Microsoft Booksh... |
edans is less than that of hybrid sedans. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 10 | HYPOTHESIS TESTING WITH TWO SAMPLES 573 98 a. H0: µd = 0 b. Ha: µd < 0 c. The random variable Xd is the average difference between hus... |
ER 11 | THE CHI-SQUARE DISTRIBUTION 579 Figure 11.3 11.2 | Goodness-of-Fit Test In this type of hypothesis test, you determine whether the data "fit" a particular distribution or not. For example, you may suspect your unknown data fit a binomial distribution. You use a chi-square test (meaning the distribution for the ... |
TE df ≠ 600 – 1 Calculate the test statistic: χ2 = 29.65 Graph: Figure 11.5 Probability statement: p-value = P(χ2 > 29.65) = 0.000006 Compare α and the p-value: • α = 0.01 • p-value = 0.000006 So, α > p-value. Make a decision: Since α > p-value, reject Ho. This means you reject the belief that the distribution for the ... |
ways right-tailed. H0: The number of hours volunteered is independent of the type of volunteer. Ha: The number of hours volunteered is dependent on the type of volunteer. The expected result are in Table 11.15. Type of Volunteer 1-3 Hours 4-6 Hours 7-9 Hours Community College Students 90.57 Four-Year College Students 1... |
table values by row. Press ENTER after each. Press 2nd QUIT. Press STAT and arrow over to TESTS. Arrow down to C:χ2-TEST. Press ENTER. You should see Observed:[A] and Expected:[B]. Arrow down to Calculate. Press ENTER. The test statistic is 3.2603 and the p-value = 0.1959. Do the procedure a second time but arrow down... |
uctory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION 601 Figure 11.10 12. State your decision. 13. State your conclusion in a complete sentence. Exponential Distribution Test to see if grocery receipts follow the exponential distribution with decay parameter 1 x¯ . 1. Us... |
following information to answer the next five exercises: A teacher predicts that the distribution of grades on the final exam will be and they are recorded in Table 11.27. Grade Proportion A B C D 0.25 0.30 0.35 0.10 Table 11.27 The actual distribution for a class of 20 is in Table 11.28. Grade Frequency A B C D 7 7 5... |
expected frequency to two decimal places. 72. A six-sided die is rolled 120 times. Fill in the expected frequency column. Then, conduct a hypothesis test to determine if the die is fair. The data in Table 11.34 are the result of the 120 rolls. Face Value Frequency Expected Frequency 1 2 3 4 5 6 Table 11.34 15 29 16 15 ... |
West skinny fries curly fries steak fries Table 11.48 70 100 20 50 60 40 20 15 10 25 30 10 92. According to Dan Lenard, an independent insurance agent in the Buffalo, N.Y. area, the following is a breakdown of the amount of life insurance purchased by males in the following age groups. He is interested in whether the ... |
lay, which distribution would you use? 120. If an additional test were done on the claim of the average delay, but 45 flights were surveyed, which distribution would you use? For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E for the chi-square solution sheet. Round expec... |
addition, all values must be greater than or equal to five. 59 a test of a single variance 61 a left-tailed test 63 H0: σ2 = 0.812; Ha: σ2 > 0.812 65 a test of a single variance 67 0.0542 69 true 71 false 73 Marital Status Percent Expected Frequency never married married widowed 31.3 56.1 2.5 divorced/separated 10.1 Ta... |
ble for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 633 12 | LINEAR REGRESSION AND CORRELATION Figure 12.1 Linear regression and correlation can help you determine if an auto mechanic’s salary is related to his work ... |
Collect data from your class (pinky finger length, in inches). The independent variable, x, is pinky finger length and the dependent variable, y, is height. For each set of data, plot the points on graph paper. Make your graph big enough and use a ruler. Then "by eye" draw a line that appears to "fit" the data. For yo... |
y. The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y. The correlation coefficient is calculated as r = nΣ(xy) − (Σx)(Σy) ⎡ ⎡ ⎣nΣx2 − (Σx)2⎤ ... |
11 data points. • Can the regression line be used for prediction? Given a third exam score (x value), can we use the line to predict the final exam score (predicted y value)? H0: ρ = 0 Ha: ρ ≠ 0 α = 0.05 • The p-value is 0.026 (from LinRegTTest on your calculator or from computer software). • The p-value, 0.026, is le... |
east squares line. They have large "errors", where the "error" or residual is the vertical distance from the line to the point. Outliers need to be examined closely. Sometimes, for some reason or another, they should not be included in the analysis of the data. It is possible that an outlier is a result of erroneous da... |
1 Example 12.13 Using this new line of best fit (based on the remaining ten data points in the third exam/final exam example (http://cnx.org/content/m47117/1.3/##element-22) ), what would a student who receives a 73 on the third exam expect to receive on the final exam? Is this the same as the prediction made using the... |
20 cars from the list and record their weights and fuel efficiency into Table 12.11. Weight Fuel Efficiency Table 12.11 2. Which variable should be the dependent variable and which should be the independent variable? Why? 3. By hand, do a scatterplot of “weight” vs. “fuel efficiency”. Plot the points on graph paper. La... |
453 1,482 3,466 6,878 11,987 16,162 20,868 Table 12.12 Adults and Adolescents only, United States 668 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 42,674 48,634 59,660 78,530 78,834 71,874 68,505 59,347 47,149 38,393 25,174 25,522 25,643 26,464 27... |
and cell phone usage in the United States. Construct a scatter plot of the data Year Poverty Rate Cellular Usage per Capita 2003 12.7 2005 12.6 2007 2009 12 12 Table 12.17 54.67 74.19 84.86 90.82 61. Does the higher cost of tuition translate into higher-paying jobs? The table lists the top ten colleges based on mid-car... |
Death from heart diseases 221 167 131 191 220 297 71 172 211 300 Table 12.25 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 679 a. Enter the data into your calculator and make a scatter pl... |
timated total cost for a next taxable estate of $1,000,000. Find the cost for $2,500,000. g. Does it appear that a line is the best way to fit the data? Why or why not? h. Are there any outliers in the data? i. Based on these results, what would be the probate fees and taxes for an estate that does not have any assets?... |
The correlation coefficient xi = –0.2752 is not less than 0.666 so we do not reject. e. page 10: 17.08 page 70: 16.23 f. There is not a significant linear correlation so it appears there is no relationship between the page and the amount of the discount. g. page 200: 14.39 h. No, using the regression equation to predi... |
othesis tests comparing averages between more than two groups, statisticians have developed a method called "Analysis of Variance" (abbreviated ANOVA). In this chapter, you will study the simplest form of ANOVA called single factor or one-way ANOVA. You will also study the F distribution, used for one-way ANOVA, and th... |
= 0.3769 MS(Factor) = SS(Factor)/(k – 1) = 2.2458/2 = 1.1229 MS(Error) = SS(Error)/(n – k) = 20.8542/7 = 2.9792 13.1 As part of an experiment to see how different types of soil cover would affect slicing tomato production, Marist College students grew tomato plants under different soil cover conditions. Groups of three... |
in inches) 24, 28, 25, 30, and 32. Do a one-way ANOVA test on the four groups. Are the heights of the bean plants different? Use the same method as shown in Example 13.4. From the class, create four groups of the same size as follows: men under 22, men at least 22, women under 22, women at least 22. Have each member of... |
is the ratio of a measure of the variation in the group means to a similar measure of the variation within the groups. If the null hypothesis is correct, then the numerator should be small compared to the denominator. A small F statistic will result, and the area under the F curve to the right will be large, representi... |
.18 16 41 22 31 State SSbetween, SSwithin, and the F statistic. 60. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the coun... |
6.4 96.8 96.3 96.9 98.4 98.6 98.1 98.6 96.7 97.7 96.7 97 98.7 98.6 98.1 98.6 Table 13.30 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 723 FL FH ML MH FL FH ML MH 97.2 97.8 97.1 97.1 98.7 9... |
Conclusion: There is insufficient evidence to conclude that the mean lap times are different. 68 a. Ha: µd = µn = µh b. At least any two of the magazines have different mean lengths. c. df(num) = 2, df(denom) = 12 d. F distribtuion e. F = 15.28 This content is available for free at http://textbookequity.org/introducto... |
roximation to the binomial, 385 Normal Distribution, 355, 394, 441, 497 normal distribution, 423, 427, 474 Normal distribution, 534 normally distributed, 370, 375, 475 null hypothesis, 470, 475, 475, 477 Numerical Variable, 46 Numerical variables, 11 O observed values, 579 One-Way ANOVA, 712 ordinal scale, 29 outcome, ... |
onrepeating, and are nonterminating: {h | h is not a rational number}. _ n ∣ m and n are integers and n ≠ 0 } m Example 5 Differentiating the Sets of Numbers Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q'). a. √ — 36 8 _ b. 3 c. √ — 7... |
d. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or 1 _ a , that, when multiplied by the original number, results in the multiplicative identity, 1. reciprocal), denoted 1 _ a = 1 a ∙ Download the OpenStax text for free at http://cnx.org/content/col11759... |
sociative Properties allow us to do when following the order of operations? Explain your answer. NUMERIC For the following exercises, simplify the given expression. 4. 10 + 2 · (5 − 3) 5. 6 ÷ 2 − (81 ÷ 32) 6. 18 + (6 − 8)3 7. −2 · [16 ÷ (8 − 4)2] 2 8. 4 − 6 + 2 · 7 9. 3(5 − 8) 10. 4 + 6 − 10 ÷ 2 11. 12 ÷ (36 ÷ 9) + 6 1... |
factors = The exponent of the answer is the product of the exponents: (x 2. In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents. (a m)n = am ∙ n Be careful to distinguish between uses of the product rule and the power rule. When u... |
= = = The power rule Simplify. The negative exponent rule The product rule Simplify. The negative exponent rule The power of a quotient rule The power of a product rule The quotient rule Simplify. The negative exponent rule Commutative and associative laws of multiplication The product rule Simplify. The product rule ... |
11. 113 ÷ 114 12. 65 · 6−7 For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents. 15. 42 · 43 ÷ 4−4 18. 106 ÷ (1010) 17. (123 · 12) 16. −2 10 612 ___ 69 19. 7−6 · 7−3 20. (33 ÷ 34) 5 For the following exercises, express the decimal in scie... |
t may change the radicand. The radical expression √ 18 can be written with a 2 in the radicand, as 3 √ — 2 and 3 √ — 2 is 4 √ — 2 , so √ 2 + 3 √ 2 = 4 √ 18 = √ 2 + √ — 2 . — — — — How To… Given a radical expression requiring addition or subtraction of square roots, solve. 1. Simplify each radical expression. 2. Add or ... |
ab2 − b √ — a 44. 3 √ — 44z + √ — 99z 45. √ — 50y8 3 _ 48. q √ 2 — 63p 49. — √ 8 ________ 1 − √ 3x — 3 _ 2 51. w √ — 3 _ 32 − w 2 √ — 50 52. √ — 108x4 + √ — 27x4 53. — 12x √ ________ 3 2 + 2 √ — 55. √ — 125n10 56. √ ____ 42q ____ 36q3 57. √ ______ 81m _ 361m2 42. — 2n 4 √ _______ 16n4 √ — 46. √ — 490bc2 50. √ ______ 20... |
distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms. How To… Given two binomials, use FOIL to simplify the expression. 1. Multiply the first terms of each binomial. 2. Multiply the outer terms of the binomials. 3. Multiply ... |
e of length 10x − 8 and one side of length 4, giving an area of A = lw = 4(10x − 8) = 40x − 32 units 2. So the region that must be subtracted has an area of 2(16) + 40x − 32 = 40x units 2. The area of the region that requires grass seed is found by subtracting 60x2 − 40x units 2. This area can also be expressed in fact... |
example. The sign of the first 2 is the same as the sign between x 3 − 23. The sign of the 2x term is opposite the sign between x 3 − 23. And the sign of the last term, 4, is always positive. x3 − 23 = (x − 2)(x2 + 2x + 4) sum and difference of cubes We can factor the sum of two cubes as We can factor the difference of... |
ltiplication would rewrite as the product 3 expression, we can multiply as we did before. 3 1 __ __ · x2 x 3 __ = x3 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 60 CHAPTER 1 PREREQUISITES How To… Given two rational expressions, divide them. 1. Rewrite as the first rational expression ... |
− 1 _ c + 1 2c _ + c + 2 __ 2c + 1 _ c + 1 REAL-WORLD APPLICATIONS 51. Brenda is placing tile on her bathroom floor. The area of the floor is 15x2 − 8x − 7 ft 2. The area of one tile is x2 − 2x + 1 ft 2. To find the number of tiles needed, simplify the rational expression: 15x2 − 8x − 7 __ . x2 − 2x + 1 44. p x _ _ − 4... |
e square root of the product ab is equal to the product of the square roots of a and b See Example 2 and Example 3. a __ is equal to the quotient of the square roots of a and b • If a and b are nonnegative, the square root of the quotient b See Example 4 and Example 5. • We can add and subtract radical expressions if t... |
is into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers—the displacement from the horizontal axis and the displacement from the vertical axis. While there is evidence that ideas similar to Descartes’ grid system existed centuries earlier, it ... |
the graph crosses the axes where we predicted it would. –5 –4 –3 –2 y = 3x − 1 321 4 5 x y 4 3 2 1 –1 –1 –2 –3 –4 Figure 11 given an equation, find the intercepts. • Find the x-intercept by setting y = 0 and solving for x. • Find the y-intercept by setting x = 0 and solving for y. Example 4 Finding the Intercepts of th... |
http://cnx.org/content/col11759/latest. SECTION 2.1 SECTION EXERCISES 85 32. (−3, 0)(−3, 4)(−3, −3) 33. Name the coordinates of the points graphed. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –5 –4 –3 –2 21 3 4 5 x A –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 B C 5 x 21 3 4 34. Name the quadrant in which the following points would b... |
rs; 2x, 3x, and 3. The LCD must contain 2x, 3x, and 3. An LCD of 6x contains all three denominators. In other words, each denominator can be divided evenly into the LCD. Next, multiply both sides of the equation by the LCD 6x. − 7 5 22 ) = ( (6x) ( ) (6x) _ _ _ 2x 3x 3 5 7 22 ) (6x) ) = ( ) − (6x) ( (6x) ( _ _ _ 3 3x 2... |
ion. If done correctly, the same final equation will be obtained. Try It #8 Given m = 4, find the equation of the line in slope-intercept form passing through the point (2, 5). Example 11 Finding the Equation of a Line Passing Through Two Given Points Find the equation of the line passing through the points (3, 4) and ... |
8. − − 59. −− 60. y =x + 61. y =x − − 62. = (,) + 63. = (,) + 64. = (,) 65. = (,) EXTENSIONS 66. y −y=mx −x x xyym 68. x −y = 70. REAL-WORLD APPLICATIONS 71. Th s 2.5 ft, fin x 67. Ax +By =C y ABCx. Th 69. − 72. fi x y p =x +y y p = x = Download the OpenStax text for free at http://cnx.org/content/col11759/late... |
lution The perimeter formula is standard: P = 2L + 2W. We have two unknown quantities, length and width. However, we can write the length in terms of the width as L = W + 3. Substitute the perimeter value and the expression for length into the formula. It is often helpful to make a sketch and label the sides as in Figu... |
ield. The length is 200 ft more than the width, and the perimeter is 1,040 ft. Find the length and width. Use the perimeter formula P = 2L + 2W. 42. Distance equals rate times time, d = rt. Find the distance Tom travels if he is moving at a rate of 55 mi/h for 3.5 h. 43. Using the formula in the previous exercise, find... |
lying because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator to write the answer in standard form a + bi. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end ... |
a string of terms separated by plus or minus signs. So, in that sense, the operation of multiplication undoes the operation of factoring. For example, expand the factored expression (x − 2)(x + 3) by multiplying the two factors together. (x − 2)(x + 3) = x2 + 3x − 2x − 6 = x2 + x − 6 The product is a quadratic expressi... |
atic by Completing the Square Solve the quadratic equation by completing the square: x 2 − 3x − 5 = 0. Solution First, move the constant term to the right side of the equal sign. x 2 − 3x = 5 1 _ of the b term and square it. Then, take 2 Add the result to both sides of the equal sign. 3 1 _ _ (−3 − 3x + ( − ) = − 3x + ... |
root. 1 __ is another way of writing √ For example, 16 2 exponents is a useful skill, as it is highly applicable in calculus. 1 __ 16 ; 8 3 — is another way of writing 3 √ — 8 . The ability to work with rational We can solve equations in which a variable is raised to a rational exponent by raising both sides of the eq... |
ion. To solve an equation such as |2x − 6| = 8, we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is 8 or −8. This leads to two different equations we can solve independently. or 2x − 6 = 8 2x = 14 x = 7 2x − 6 = −8 2x = −2 x = −1 Knowing how to solve problems involving... |
both and and or type. • ole absolute alue inequalities. 2.7 LINEAR INEQUALITIES AND ABSOLUTE VALUE INEQUALITIES It is not easy to make the honor role at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could ... |
all x -values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 2.7 LINEAR INEQUALITIES AND ABSOLUTE VALUE INEQUALITIES 147 There are two basic approaches... |
s isolated so that the square root of both sides of the equation can be taken to solve for x volume in cubic units, the volume measurement includes length, width, and depth: V = LWH x-axis the common name of the horizontal axis on a coordinate plane; a number line increasing from left to right x-coordinate the first co... |
10. y = x + 4 2 11. 4x − 3y = 6 LINEAR EQUATIONS IN ONE VARIABLE For the following exercises, solve for x. 12. 5x + 2 = 7x − 8 13. 3(x + 2) − 10 = x + 4 14. 7x − 3 = 5 15. 12 − 5(x + 1) = 2x − 5 16. 2x 21 _ 4 For the following exercises, solve for x. State all x-values that are excluded from the solution set. 17. x _ ... |
ot functions. Relation is a Function Outputs Inputs Relation is a Function Inputs Outputs Relation is NOT a Function Inputs p q Outputs x y z (a) (b) (c) Figure 1 ( a ) This relationship is a function because each input is associated with a single output. Note that input q and r both give output n. ( b ) This relations... |
ing function notation. The function represented by Table 6 can be represented by writing Similarly, the statements f (2) = 1, f (5) = 3, and f (8) = 6 g (−3) = 5, g (0) = 1, and g (4) = 5 represent the function in table Table 7. Table 8 cannot be expressed in a similar way because it does not represent a function. Try ... |
point. The point has coordinates (2, 1), so f (2) = 1. See Figure 7. f (x) 7 6 5 4 3 2 1 –1 –1 2 – – 3 –5 –4 –3 –2 (2, 1) f (2) = 1 21 3 4 5 Figure 7 b. To solve f (x) = 4, we find the output value 4 on the vertical axis. Moving horizontally along the line y = 4, we locate two points of the curve with output value 4: (... |
put and the a function? output of a function? 3. Why does the vertical line test tell us whether the 4. How can you determine if a relation is a one-to-one graph of a relation represents a function? function? 5. Why does the horizontal line test tell us whether the graph of a function is one-to-one? ALGEBRAIC For the f... |
. Brackets, [or], are used to indicate that an endpoint value is included, called inclusive. See Figure 3 for a summary of interval notation. Inequality Interval Notation Graph on Number Line a, ∞) (−∞, a) [a, ∞) (−∞, a] (a, b) [a, b) (a, b] [a, b Figure 3 Description x is greater than a x is less than a x is greater t... |
–2 –1 f –1 –2 – 3 – 4 –5 1 2 3 4 5 x Range Figure 10 Example 7 Finding Domain and Range from a Graph of Oil Production Find the domain and range of the function f whose graph is shown in Figure 11. Alaska Crude Oil Production Th 2200 2000 1800 1600 1400 1200 1000 800 600 400 200 0 1975 1980 1985 1990 1995 2000 2005 Fi... |
an 2, we use the first formula. C(1.5) = $25 To fi nd the cost of using 4 gigabytes of data, C(4), we see that our input of 4 is greater than 2, so we use the second formula. C(4) = 25 + 10(4 − 2) = $45 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 3.2 DOMAIN AND RANGE 191 Analy... |
years ≈ 0.196 dollars per year On average, the price of gas increased by about 19.6¢ each year. Other examples of rates of change include: • A population of rats increasing by 40 rats per week • A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes) • A car driving 27 miles p... |
h Given the function p(t) in Figure 6, identify the intervals on which the function appears to be increasing. p 5 4 3 2 1 –1 1 2 3 4 5 6 t –1 –2 Figure 6 Solution We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it s... |
reference to a year and write T = h + w Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of alg... |
the input and output values, but this time, from the x- and y-axes of the graphs. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 214 CHAPTER 3 FUNCTIONS How To… Given a composite function and graphs of its individual functions, evaluate it using the information provided by the graphs. 1... |
gx Table 3 58. fg 62. ff 59. fg 63. ff 60. gf 64. gg 61. gf 65. gg fgTable 4 − − x f (x) g(x) − − − Table 4 − − − 66. f"∘"g 69. g"∘"f 67. f"∘"g 70. g"∘"g 68. g"∘"f 71. f"∘"f o fif ggf 72. fx=x+gx=−x 73. fx=x+gx=−x 74. fx=√ — x+gx=−x 75. fx= gx=x+ _ x +" f x=x+gx=x +fi 76. fg 77. fgx 78. g(f− 79. g"∘"g x Download the Op... |
hat the output values of g are the same as the output value of f when the input value is 3 less than the original value. For example, we know that f (2) = 1. To get the same output from the function g, we will need an input value that is 3 larger. We input a value that is 3 larger for g (x) because the function takes 3... |
function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the h(x) values stay the same as the f (x) values. See Table 8. x h(x) –2 1 –4 3 Table 8 –6 7 –8 11 Try It #6 A function f (x) is given as Table 9. Create a table for the functions below. x f(x) −2 5 0 ... |
759/latest. SECTION 3.5 TRANSFORMATION OF FUNCTIONS 237 Solution When trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that g(2) = 2. With the basic cubic function at the same input, f (2) = 23 = 8. the outputs of the fun... |
–5 21 3 4 5 6 x Figure 29 Last, we vertically shift down by 3 to complete our sketch, as indicated by the −3 on the outside of the function. See Figure 30. f (x) –6 –5 –4 –3 –2 5 4 3 2 1 –1 –1 –2 –3 –4 –5 21 3 4 5 6 x Figure 30 Access this online resource for additional instruction and practice with transformation of ... |
A and B, an equation of the form | A |#= B, with B ≥ 0, will have solutions when A = B or A = −B. If B <#0, the equation | A |#= B has no solution. How To… Given the formula for an absolute value function, find the horizontal intercepts of its graph. 1. Isolate the absolute value term. 2. Use | A |#= B to write A = B ... |
terchange the input and output of each from the graph of the function y = 4 coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function. inverse function For any one-to-one function f (x) = y, a function f −1 (x) is an inverse function of f if f −1(y) = x. This can... |
al function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the horizontal extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we fi... |
ion that shifts a function’s graph left or right by adding a positive or negative constant to the input horizontal stretch a transformation that stretches a function’s graph horizontally by multiplying the input by a constant 0 < b < 1 increasing function a function is increasing in some open interval if f (b) > f (a) ... |
which the reflections are applied does not affect the final graph. See Example 9. • A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See Example 10. • A function presented as an equation can be reflected by applying transformation... |
rmine whether the functions are even, odd, or neither. 5 _ 15. f (x) = − x2 + 9x 6 1_ 17. f (x) = x 5 _ 16. f (x) = − x 3 + 9x 5 18. Graph the absolute value function f (x) = −2| x − 1 | + 3. For the following exercises, find the inverse of the function. 19. f (x) = 3x − 5 20. f (x) = 4 _____ x + 7 For the following ex... |
a positive slope slants upward from left to right as in Figure 5(a). For a decreasing function, the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in Figure 5(b). If the function is constant, the output values are the same f... |
ich he incurs a fixed cost of $1,250 per month for the overhead, which includes his office rent. His production costs are $37.50 per item. Write a linear function C where C(x) is the cost for x items produced in a given month. Solution The fixed cost is present every month, $1,250. The costs that can vary include the c... |
pe is positive, we know the graph will slant upward from left to right. The y-intercept is 2 the point on the graph when x = 0. The graph crosses the y-axis at (0, 1). Now we know the slope and the y-intercept. rise _ We can begin graphing by plotting the point (0, 1). We know that the slope is rise over run, m = run .... |
, are the only examples of linear functions with no x-intercept. For example, y = 5 is a horizontal line 5 units above the x-axis. This function has no x-intercepts, as shown in Figure 21. y y = 5 21 3 4 5 x –5 –4 –3 –2 5 4 3 2 1 –1–1 –2 –3 –4 –5 Figure 21 x-intercept The x-intercept of the function is value of x when ... |
s through a given point. Suppose then we want to write the equation of a line that is perpendicular to f (x) and passes through the point (4, 0). We already know that the slope is − #1 __ . Now we can use the point to find the y-intercept by substituting 2 the given values into the slope-intercept form of a line and so... |
86. 21 3 4 5 6 x –6 –5 –4 –3 –2 y 6 5 4 3 2 1 –1–1 –2 –3 –4 –5 –6 87. 21 3 4 5 6 x –6 –5 –4 –3 –2 y 6 5 4 3 2 1 –1–1 –2 –3 –4 –5 –6 78. h(x) =# #1 __ x + 2 3 81. p(t) = −2 + 3t 84. r(x) = 4 88. 21 3 4 5 6 x –5 –4 –3 –2 y 5 4 3 2 1 –1–1 –2 –3 –4 –5 21 3 4 5 x Download the OpenStax text for free at http://cnx.org/conten... |
most no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesn’t make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before she saved $3,500, but the scenario discussed poses the question once she saved $3,500 because... |
Should I draw diagrams when given information based on a geometric shape? Yes. Sketch the figure and label the quantities and unknowns on the sketch. Example 3 Using a Diagram to Model Distance Between Cities There is a straight road leading from the town of Westborough to Agritown 30 miles east and 10 miles north. Par... |
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