text stringlengths 235 3.08k |
|---|
How To… Given two complex numbers, find the sum or difference. 1. Identify the real and imaginary parts of each number. 2. Add or subtract the real parts. 3. Add or subtract the imaginary parts. Example 3 Adding and Subtracting Complex Numbers Add or subtract as indicated. a. (3 − 4i) + (2 + 5i) b. (−5 + 7i) − (−11 + ... |
method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The difference with complex numbers is that when we get a squared term, i2, it equals −1. (a + bi)(c + di) = ac + adi + bci + bdi2 = ac + adi + bci − bd = (ac − bd) + (ad + bc)i i2... |
relationship: The complex conjugate of a + bi is a − bi, and the complex conjugate of a − bi is a + bi. Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another. Suppose we want to divide c + di by a + bi, where neither a nor b equals z... |
— 5. 1 _ i. This can be written simply as 2 Analysis Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number, we can si... |
1 ċ i = i We can see that when we get to the fifth power of i, it is equal to the first power. As we continue to multiply i by increasing powers, we will see a cycle of four. Let’s examine the next four powers of i. i6 = i5 ċ i = i ċ i = i2 = −1 i7 = i6 ċ i = i2 ċ i = i3 = −i i8 = i7 ċ i = i3 ċ i = i4 = 1 i9 = i8 ċ i ... |
org/l/multcompconj) • Raising i to Powers (http://openstaxcollege.org/l/raisingi) Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 118 CHAPTER 2 EQUATIONS AND INEQUALITIES 2.4 SECTION EXERCISES VERBAL 1. Explain how to add complex numbers. 2. What is the basic principle in multiplication o... |
6 + 4i_ i 3 + 4i_ 2 2 − 3i _ 4 + 3i 32. 35. √ — −9 + 3 √ — −16 36. − √ — −4 − 4 √ — −25 39. i8 40. i15 29. 6 − 2i_ 3 3 + 4i_ 2 − i 33. 37. — 2 + √ −12 __________ 2 41. i22 30. 34. −5 + 3i_ 2i 2 + 3i _ 2 − 3i 38. — 4 + √ −20 __________ 2 TECHNOLOGY For the following exercises, use a calculator to help answer the questi... |
In this section you will: • Solve quadratic equations by factoring. • Solve quadratic equations by the square root property. • Solve quadratic equations by using the quadratic formula. 2.5 QUADRATIC EQUATIONS Figure 1 The computer monitor on the left in Figure 1 is a 23.6-inch model and the one on the right is a 27-in... |
the leading coefficient, whether it is 1 or another integer. We will look at both situations; but first, we want to confirm that the equation is written in standard form, ax2 + bx + c = 0, where a, b, and c are real numbers, and a ≠ 0. The equation x2 + x − 6 = 0 is in standard form. We can use the zero-product proper... |
equals −6 and whose sum equals 1. Begin by looking at the possible factors of −6. 1 ċ (−6) (−6) ċ 1 2 ċ (−3) 3 ċ (−2) The last pair, 3 ċ (−2) sums to 1, so these are the numbers. Note that only one pair of numbers will work. Then, write the factors. (x − 2)(x + 3) = 0 To solve this equation, we use the zero-product pr... |
= 0. Solution Recognizing that the equation represents the difference of squares, we can write the two factors by taking the square root of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other. Solve using the zero-factor property. x 2 − 9 = 0 (x − 3)(x + 3) = 0 (x −... |
property. (4x + 3)(x + 3) = 0 (4x + 3x + 3) = 0 x = −3 3 _ and −3. See Figure 3. The solutions are − 4 2 + 15x + 9 = 0 4x –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 21 3 4 5 x Figure 3 Try It #4 Solve using factoring by grouping: 12x 2 + 11x + 2 = 0. Example 5 Solving a Higher Degree Quadratic Equation by Factoring Sol... |
x = ± √ — k where k is a nonzero real number. How To… Given a quadratic equation with an x 2 term but no x term, use the square root property to solve it. 1. Isolate the x 2 term on one side of the equal sign. 2. Take the square root of both sides of the equation, putting a ± sign before the expression on the side opp... |
Multiply the b term by 2 x 2 + 4x = −1 1 _ (4) = 2 2 22 = 4 1 b ) 3. Add ( _ 2 2 to both sides of the equal sign and simplify the right side. We have x 2 + 4x + 4 = −1 + 4 x 2 + 4x + 4 = 3 4. The left side of the equation can now be factored as a perfect square. 5. Use the square root property and solve. x 2 + 4x + 4 ... |
if it is negative, we can multiply the equation by −1 and obtain a positive a. Given ax 2 + bx + c = 0, a ≠ 0, we will complete the square as follows: 1. First, move the constant term to the right side of the equal sign: ax 2 + bx = −c 2. As we want the leading coefficient to equal 1, divide through by a: b 1 1 ) of t... |
quadratic equation: x 2 + 5x + 1 = 0. Solution Identify the coefficients: a = 1, b = 5, c = 1. Then use the quadratic formula. x = — −(5) ± √ (5)2 − 4(1)(1) ___ 2(1) — = −5 ± √ 25 − 4 __ 2 — = 21 __ −5 ± √ 2 Example 10 Solving a Quadratic Equation with the Quadratic Formula Use the quadratic formula to solve x 2 + x +... |
is the expression under the radical in the quadratic formula: b2 − 4ac. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect. Example 11 Using the Discriminant to Find the Nature of the Solutions to a Quadratic Equation Use the discriminant to find the nat... |
side. The Pythagorean Theorem is given as a2 + b2 = c2 where a and b refer to the legs of a right triangle adjacent to the 90° angle, and c refers to the hypotenuse, as shown in Figure 4. b c a Figure 4 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 128 CHAPTER 2 EQUATIONS AND INEQUALIT... |
x += 13. x = 17. x +x = 19. x= 23. x += 20. x = 24. x −= 21. x −= 22. x −= 25. x−x −= 28. x + = x − 29. +z =z 26. x −x −= 30. p+p −= 27. x −x = 31. x −x −= 32. x−x += 36. x−x −= 33. x +x += 37. x −x −= 35. x −x += 34. x +x −= 38. x +x += 42. x +x += 46. ++= TECHNOLOGY 39. x +x = x − x= 43. + += 47. 40. x−x −= 44. ++=... |
6 OTHER TYPES OF EQUATIONS We have solved linear equations, rational equations, and quadratic equations using several methods. However, there are many other types of equations, and we will investigate a few more types in this section. We will look at equations involving rational exponents, polynomial equations, radical... |
to remove the exponent on x is by raising both sides of the equation to a power that is the reciprocal 5 4 _ _., which is of 5 4 Try It #2 3 _ = 125. Solve the equation x 2 5 _ = 32 32) 5 4 x = (2)4 The fifth root of 32 is 2. = 16 Example 3 Solving an Equation Involving Rational Exponents and Factoring 3 1 _ _ = x. So... |
n is an expression of the type anx n + an − 1x n − 1 + ċ ċ ċ + a2x 2 + a1 x + a0 where n is a positive integer and an, …, a0 are real numbers and an ≠ 0. Setting the polynomial equal to zero gives a polynomial equation. The total number of solutions (real and complex) to a polynomial equation is equal to the highest e... |
terms. If the factors in the parentheses are identical, we can continue the process and solve, unless more factoring is suggested. x 3 + x 2 − 9x − 9 = 0 x 2(x + 1) − 9(x + 1) = 0 (x 2 − 9)(x + 1) = 0 The grouping process ends here, as we can factor x 2 − 9 using the difference of squares formula. (x 2 − 9)(x + 1) = 0... |
sign. Put all remaining terms on the other side. 2. If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of the equation to the third power. In other words, for an nth root radical, raise both sides to the nth power. Doing so eliminates the radical symbol... |
� — x − 2 from both sides. Square both sides. Use the perfect square formula to expand the right side: (a − b)2 = a2 −2ab + b2x − 2) x − 2 — 2x + 3 = (4)2 − 2(4) √ 2x + 3 = 16 − 8 √ 2x + 3 = 14 + x − 8 √ — x − 11 = −8 √ (x − 11)2 = ( − ) Combine like terms. Isolate the second radical. Square both sides. x 2 − 22x + 121... |
a specified distance from a given reference point. absolute value equations The absolute value of x is written as |x|. It has the following properties: If x ≥ 0, then |x| = x. If x < 0, then |x| = −x. For real numbers A and B, an equation of the form |A| = B, with B ≥ 0, will have solutions when A = B or A = −B. If B ... |
so we have to write only one equation. −5x + 10 = 0 −5x = −10 x = 2 There is one solution: 2. Try It #7 Solve the absolute value equation: |1 − 4x| + 8 = 13. Solving Other Types of Equations There are many other types of equations in addition to the ones we have discussed so far. We will see more of them throughout th... |
− 1 = 0. Solution This equation fits the main criteria, that the power on the leading term is double the power on the middle term. Next, we will make a substitution for the variable term in the middle. Let u = x 2. Rewrite the equation in u. Now solve the quadratic. 3u2 − 2u − 1 = 0 3u2 − 2u − 1 = 0 (3u + 1)(u − 1) = ... |
Rational Equation Leading to a Quadratic Solve the following rational equation: −4x 8 _. x2 − 1 Solution We want all denominators in factored form to find the LCD. Two of the denominators cannot be factored further. However, x2 −1 = (x + 1)(x − 1). Then, the LCD is (x + 1)(x − 1). Next, we multiply the whole equation ... |
5| = −7 has no solutions. ALGEBRAIC For the following exercises, solve the rational exponent equation. Use factoring where necessary. 2 _ = 16 6. x 3 2 _ = 4 10. (x + 1) 3 3 _ = 27 7 − 5x 11 − 4x − 3x 12. (x − 1) 4 For the following exercises, solve the following polynomial equations by grouping and factoring. 13. x 3... |
= 0 39. (x 2 − 1)2 + (x 2 − 1) − 12 = 0 EXTENSIONS For the following exercises, solve for the unknown variable. 42. x−2 − x−1 − 12 = 0 — |x|2 = x 43. √ 44. t 25 − t 5 + 1 = 0 45. |x 2 + 2x − 36| = 12 REAL-WORLD APPLICATIONS For the following exercises, use the model for the period of a pendulum, T, such that T = 2π √ ... |
. We can use a number line as shown in Figure 2. The blue ray begins at x = 4 and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4. 5 6 Figure 2 We can use set-builder notation: {x|x ≥ 4}, which translates to “all real nu... |
a and b, including a All real numbers between a and b, including b All real numbers between a and b, including a and b {x| x ≤ b} {x| a ≤ x < b} {x| a < x ≤ b} {x| a ≤ x ≤ b} (−∞, b] [a, b) (a, b] [a, b] All real numbers less than a or greater than b {x| x < a and x > b} (−∞, a) ∪ (b, ∞) All real numbers {x| x is all ... |
c. Multiplication Property If a < b and c > 0, then ac < bc. If a < b and c < 0, then ac > bc. These properties also apply to a ≤ b, a > b, and a ≥ b. Example 3 Demonstrating the Addition Property Illustrate the addition property for inequalities by solving each of the following: a. x − 15 < 4 b. 6 ≥ x − 1 c. x + 7 > ... |
to solving an equation up until the last step. 13 − 7x ≥ 10x − 4 13 − 17x ≥ −4 −17x ≥ −17 x ≤ 1 Move variable terms to one side of the inequality. Isolate the variable term. Dividing both sides by −17 reverses the inequality. The solution set is given by the interval (−∞, 1], or all real numbers less than and includin... |
2 The second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time. 3 ≤ 2x + 2 < 6 1 ≤ 2x < 4 1 _ ≤ x < 2 2 Isolate the variable term, and subtract 2 from all three parts. Divide through all three parts by 2. 1 We get the same solution: [, 2 ). _ 2 Try It... |
UTE VALUE INEQUALITIES 147 There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solu... |
an Absolute Value Inequality Solve |x − 1| ≤ 3. Solution |x − 1| ≤ 3 −3 ≤ x − 1 ≤ 3 −2 ≤ x ≤ 4 [−2, 4] Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 148 CHAPTER 2 EQUATIONS AND INEQUALITIES Example 11 Using a Graphical Approach to Solve Absolute Value Inequalities 1 _ Given the equatio... |
Inequalities (http://openstaxcollege.org/l/compndineqs) • Absolute Value Inequalities (http://openstaxcollege.org/l/absvalineqs) Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 2.7 SECTION EXERCISES 149 2.7 SECTION EXERCISES VERBAL 1. 2. −x > x<− x +<x + < 3. 4. x +>x + > 5. y =|... |
x + 50. x +>x + 51. x − x +> 52. x +< x + NUMERIC 53. x|−< x < 54. x|x≥ 55. x|x< 56. x| x 57. −∞ 58. ∞ 59. − 60. −∪∞ 61. − − 62. − − 63. TECHNOLOGY fts 1 Y2 =MATHNum1:abs( 2 nd CALC 5:intersection1st curve, enter, 2 nd curve, enter, guess, enter x-fi 64. |x +|−< 65. − |x +|< 66. |x +|−> 67. |x −|< 68. |x +|≥ EXTENSIONS... |
under the radical in the quadratic formula that indicates the nature of the solutions, real or complex, rational or irrational, single or double roots. distance formula a formula that can be used to find the length of a line segment if the endpoints are known equation in two variables a mathematical statement, typical... |
second-degree polynomial; can be solved using multiple methods quadratic formula a formula that will solve all quadratic equations radical equation an equation containing at least one radical term where the variable is part of the radicand rational equation an equation consisting of a fraction of polynomials slope the... |
and is used to find the length of a line segment. See Example 5 and Example 6. • The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the x-coordinates and the sum of the y-coordinates of the endpoints by 2. See Example 7 and Example 8. 2.2 Linear Equations in One Varia... |
• Many geometry problems are solved using the perimeter formula P = 2L + 2W, the area formula A = LW, or the volume formula V = LWH. See Example 4, Example 5, and Example 6. 2.4 Complex Numbers • The square root of any negative number can be written as a multiple of i. See Example 1. • To plot a complex number, we use... |
ALITIES • The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See Example 12. 2.6 Other Types of Equations • Rational exponents ... |
equal to a negative value. See Example 9 and Example 10. • Absolute value inequalities can also be solved by graphing. At least we can check the algebraic solutions by graphing, as we cannot depend on a visual for a precise solution. See Example 11. Download the OpenStax text for free at http://cnx.org/content/col1175... |
). 1 _ 20. Passes through the point (−3, 4) and has a slope of −. 3 21. Passes through the point (−3, 4) and is parallel to 22. Passes through these two points: (5, 1),(5, 7). 2 _ the graph y = x + 5. 3 MODELS AND APPLICATIONS For the following exercises, write and solve an equation to answer each question. 23. The num... |
property. 44. x2 = 49 45. (x − 4)2 = 36 For the following exercises, solve the quadratic equation by completing the square. 46. x2 + 8x − 5 = 0 47. 4x2 + 2x − 1 = 0 For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No real solution. 48. 2x2 −... |
y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y-values of the lines. See the interval where the inequality is true. x + 3 < 3x − 4 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 158 CHAPTER 2 EQUATIONS AND INEQU... |
(5 − 3i). — −4 + 3 √ 19. Simplify: √ 4 − i ______. 2 + 3i 21. Divide: 23. Solve: (3x − 1)2 − 1 = 24. 25. Solve: 4x 2 − 4x − 1 = 0 27. Solve: 2 + √ — 12 − 2x = x 22. Solve this quadratic equation and write the two complex roots in a + bi form: x 2 − 4x + 7 = 0. 24. Solve: x 2 − 6x = 13. 26. Solve 28. Solve: (x − 1) 3 Fo... |
there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties. 159 Download the OpenStax text for free at http://cnx.org/content... |
}. Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as {(odd, 1), (even, 2), (odd, 3), (even, 4), (odd, 5)} Notice that each element in the domain, {even, odd} is not paired with exactly one element in the range, {1, 2, 3, 4, 5}. For ... |
........................ 1.49 Jelly Donut.............................. 1.99 Chocolate Donut.......................... 1.99 Figure 2 Solution a. Let’s begin by considering the input as the items on the menu. The output values are then the prices. See Figure 2. Each item on the menu has only one price, so the price is a... |
a specific grade-point average. In the grading system given, there is a range of percent grades that correspond to the same grade-point average. For example, students who receive a grade-point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function... |
1 http://www.baseball-almanac.com/legendary/lisn100.shtml. Accessed 3/24/2014. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 3.1 FUNCTIONS AND FUNCTION NOTATION 163 Example 3 Using Function Notation for Days in a Month Use function notation to represent a function whose input i... |
we like to maintain a distinction between a function such as f, which is a rule or procedure, and the output y we get by applying f to a particular input x. This is why we usually use notation such as y = f (x), P = W(d), and so on. Representing Functions Using Tables A common method of representing functions is in th... |
Output Input Output 2 5 8 1 3 6 −3 0 4 5 1 5 Table 6 Table 7 0 2 4 1 5 5 Table 8 Solution Table 6 and Table 7 define functions. In both, each input value corresponds to exactly one output value. Table 8 does not define a function because the input value of 5 corresponds to two different output values. When a table rep... |
input value is a number, 2, we can use simple algebra to simplify. f (2) = 22 + 3(2. In this case, the input value is a letter so we cannot simplify the answer any further. c. With an input value of a + h, we must use the distributive property. f (a) = a2 + 3a − 4 f (a + h) = (a + h)2 + 3(a + h) − 4 = a2 + 2ah + h2 + ... |
) = 0 Factor. If (p + 3)(p − 1) = 0, either (p + 3) = 0 or (p − 1) = 0 (or both of them equal 0). We will set each factor equal to 0 and solve for p in each case. (p + 3) = 0, p = −3 (p − 1) = 0, p = 1 This gives us two solutions. The output h(p) = 3 when the input is either p = 1 or p = −3. We can also verify by graph... |
from both sides. p = 12 − 2n _______ 6 Divide both sides by 6 and simplify. p = 12 __ − 6 2n ___ 6 Therefore, p as a function of n is written as 1 __ p = 2 − n 3 1 __ p = f (n) = 2 − n 3 Example 10 Expressing the Equation of a Circle as a Function Does the equation x 2 + y 2 = 1 represent a function with x as input an... |
dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours. The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See Table 10.[2] Pet Puppy Adult dog Cat Goldfish Beta fish Memory span in hours 0.008 ... |
for free at http://cnx.org/content/col11759/latest. SECTION 3.1 FUNCTIONS AND FUNCTION NOTATION 169 Try It #7 Using Table 11, evaluate g(1). Finding Function Values from a Graph Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find ... |
stock chart shown in Figure 1 at the beginning of this chapter, the stock price was $1,000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1,000. However, some functions have only one input value for each output value, as well as having only o... |
the bank account number? Try It #10 a. If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade? b. If so, is the function one-to-one? Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 3.1 FUNCTIONS AND FUNCTION NO... |
. 172 CHAPTER 3 FUNCTIONS Example 14 Applying the Vertical Line Test Which of the graphs in Figure 11 represent( s ) a function y = f (x)? f (x) (a) y 5 4 3 2 1 f (x) x 4 5 6 7 8 9 10 11 12 x x (b) Figure 11 (c) Solution If any vertical line intersects a graph more than once, the relation represented by the graph is no... |
Identifying Basic Toolkit Functions In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When workin... |
f (x) x –2 –1 –0.5 0.5 1 2 x –2 –1 –0.5 0.5 1 2 x 0 1 4 f(x) –0.5 –1 –2 2 1 0.5 f(x) 0.25 1 4 4 1 0.25 f(x) 0 1 2 x –1 f(x) –1 –0.125 –0.5 0 0.125 1 0 0.5 1 x x x x Table 13 Access the following online resources for additional instruction and practice with functions. • Determine if a Relation is a Function (http://ope... |
y = 20. x = y 3 23. x = ± √ — 1 − y 26. y 3 = x 2 12. 2x + y 2 = 6 15. x = 3y + 5 _ 7y − 1 18. x 2 + y 2 = 9 21. y = x 3 24. y = ± √ — 1 − x 13. y = −2x 2 + 40x 16. x = √ — 1 − y 2 19. 2xy =#1 22. y = √ — 1 − x 2 25. y 2 = x 2 For the following exercises, evaluate the function f at the indicated values f (−3), f (2), ... |
). c. Solve f (t) = 2. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 3.1 SECTION EXERCISES 177 GRAPHICAL For the following exercises, use the vertical line test to determine which graphs show relations that are functions. 40. y 41. y 42. y 43. y 46. y 49. y x x x x 44. y 47. y 5... |
2, −2),(−3, −3)} 61. {(3, 4),(4, 5),(5, 6)} 62. {(2, 5),(7, 11),(15, 8),(7, 9)} For the following exercises, determine if the relation represented in table form represents y as a function of x. 63. x y 5 3 10 8 15 14 64. x y 5 3 10 8 15 8 65. x y 5 3 10 8 10 14 For the following exercises, use the function f represente... |
√ viewing window. Show each graph. — x on the given viewing window. Determine the corresponding range for each 82. [0, 0.01] 83. [0, 100] 84. [0, 10,000] For the following exercises, graph y = viewing window. Show each graph. 85. [−0.001, 0.001] — 3 √ x on the given viewing window. Determine the corresponding range fo... |
the mood for a scary movie, you may want to check out one of the five most popular horror movies of all time—I am Legend, Hannibal, The Ring, The Grudge, and The Conjuring. Figure 1 shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in gener... |
interval notation, we use a square bracket [when the set includes the endpoint and a parenthesis (to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100... |
function: {(2, 10), (3, 10), (4, 20), (5, 30), (6, 40)}. Solution First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs. {2, 3, 4, 5, 6} Try It #... |
the denominator equal to 0 and solve for x. 2 − x = 0 −x = −2 x = 2 Now, we will exclude 2 from the domain. The answers are all real numbers where x < 2 or x > 2 as shown in Figure 4. We can use a symbol known as the union, ∪, to combine the two sets. In interval notation, we write the solution: (−∞, 2) ∪ (2, ∞). –3 –... |
on an attendance chart), in such cases the domain and range have no elements in common. has the set of all positive real numbers as its domain but the set of all Using Notations to Specify Domain and Range In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use ine... |
. For sets of real numbers on intervals, another example of a union is {x | | x | ≥ 3} = (−∞, −3] ∪ [3, ∞) set-builder notation and interval notation Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form {x | statement about x} which is read as, “the set of... |
content/col11759/latest. SECTION 3.2 DOMAIN AND RANGE 185 Try It #5 Given this figure, specify the graphed set in a. words b. set-builder notation c. interval notation Finding Domain and Range from Graphs –5 –4 –3 –2 –1 0 1 2 3 4 5 Figure 7 Another way to identify the domain and range of functions is by using graphs. B... |
Alaska Crude Oil Production Th 2200 2000 1800 1600 1400 1200 1000 800 600 400 200 0 1975 1980 1985 1990 1995 2000 2005 Figure 11 (credit: modification of work by the U.S. Energy Information Administration) [4] Solution The input quantity along the horizontal axis is “years,” which we represent with the variable t for t... |
17 For the identity function f(x) = x, there is no restriction on x. Both the domain and range are the set of all real numbers. Domain: (−∞, ∞) Range: (−∞, ∞) x Domain: (−∞, ∞) Range: [0, ∞) x For the absolute value function f(x) = ∣ x ∣, there is no restriction on x. However, because absolute value is defined as a di... |
: [0, ∞) Range: [0, ∞) x Domain: (−∞, ∞) Range: (−∞, ∞) x — x, we cannot take the For the square root function f(x) = √ square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number x is defined to be positive, even thoug... |
square root of a negative number, so the value inside the radical must be nonnegative. The domain of f (x) is [−4, ∞). x + 4 ≥ 0 when x ≥ −4 We then find the range. We know that f (−4) = 0, and the function value increases as x increases without any upper limit. We conclude that the range of f is [0, ∞). Analysis Figu... |
) if S > $10,000. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 190 CHAPTER 3 FUNCTIONS piecewise function A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all ... |
, C(1.5), we first look to see which part of the domain our input falls in. Because 1.5 is less than 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 OpenSt... |
if 1 < x ≤ 2; ( c) f (x ) = x if x > 2 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 192 CHAPTER 3 FUNCTIONS Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure 26. f (x) 5 4 3 2 1 – Figure 26 Analysis Note that the graph does pas... |
f (x) = −2x(x − 1)(x − 2) 7. f (x) = 5 − 2x2 8. f (x) = 3 √ — x − 2 9. f (x) = 3 − √ — 6 − 2x 10. f (x) = √ — 4 − 3x 11. f (x) = √ — x 2 + 4 12. f (x) = # 3 √ — 1 − 2x 13. f (x) = # 3 √ — x − 1 15. f (x) = #3x + 1 ______ 4x + 2 18. f (x) = # 1 ________ x2 − x − 6 16. f (x) = # — √ x + 4 _______ x − 4 19. f (x) = # 2x3... |
33. 36. CHAPTER 3 FUNCTIONS y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 y 5 4 3 2 1 y –1 –1 –2 –3 –4 –5 10 8 6 4 2 –2 –2 –4 –6 –8 –10 –5 –4 –3 –2 –5 –4 –3 –2 –10 –8 –6 –4 31. 21 3 4 5 x –5 –4 –3 –2 34. 21 3 4 5 x –5 –4 –3 –2 37. 42 6 8 10 x –10 –8 –6 –4 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 y 5 4 3 2 1 y –1 –1 –2 –3 –4 –5 10 8 6 4 2 –2 –2 ... |
content/col11759/latest. SECTION 3.2 SECTION EXERCISES 195 NUMERIC For the following exercises, given each function f, evaluate f (−3), f (−2), f (−1), and f (0). { 46. f (x) =# x + 1 if x < −2 −2x − 3 if x ≥ −2 { 47. f (x) =# 1 if x ≤ −3 0 if x > −3 { 48. f (x) =# −2x 2 + 3 if x ≤ −1 if x > −1 5x − 7 For the following... |
given by the function h(t) = −16t 2 + 96t. What is the domain of the function? What does the domain mean in the context of the problem? 61. The cost in dollars of making x items is given by the function C(x) = 10x + 500. a. The fixed cost is determined when zero items are produced. Find the fixed cost for this item. b... |
__ Change in input ∆y_ ∆x y2 − y1 _ x2 − x1 f (x2) − f (x1) __ − x1 = = = x2 The Greek letter ∆ (delta) signifies the change in a quantity; we read the ratio as “delta-y over delta-x” or “the change in y divided by the change in x.” Occasionally we write ∆ f instead of ∆y, which still represents the change in the func... |
and x2. 1. Calculate the difference y2 − y1 = ∆y. 2. Calculate the difference x2 − x1 = ∆x. ∆y _. 3. Find the ratio ∆x Example 1 Computing an Average Rate of Change Using the data in Table 1, find the average rate of change of the price of gasoline between 2007 and 2009. Solution In 2007, the price of gasoline was $2.... |
that the order we choose is very important. If, for example, we use answer. Decide which point will be 1 and which point will be 2, and keep the coordinates fixed as ( x1, y1, we will not get the correct ). ) and ( x2, y2 y2 − y1 _ x1 − x2 Example 3 Computing Average Rate of Change from a Table After picking up a frie... |
of change of force if the distance between the particles is increased from 2 cm to 6 cm. Solution We are computing the average rate of change of F (d) = 2 __ d 2 on the interval [2, 6]. Average rate of change = = F(6) − F(2) __ 6 − 2 2 2 _ _ 62 − 22 _ 6 − 2 2 −# #2 __ __ 36 4 _ 4 − #16 __ 36 _ 4 = − #1 __ 9 = = Simpli... |
shows examples of increasing and decreasing intervals on a function. Increasing –5 –4 –3 –2 f(x) 20 16 12 8 4 –1 –4 –8 –12 –16 –20 Decreasing 21 3 4 5 x Increasing f(b) > f(a) where b > a f(b) < f(a) where b > a f(b) > f(a) where b > a Figure 3 The function f(x ) = x 3 − 12x is increasing on (−∞, −2) ∪ (2, ∞) and is d... |
OF CHANGE AND BEHAVIOR OF GRAPHS 201 To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than... |
FUNCTIONS Analysis Notice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at t = 1, t = 3, and t = 4. These points are the local extrema (two minima and a maximum). Example 8 Finding Local Extrema from a Graph x 2 __ _... |
ima and minima. y 10 8 6 4 2 –1 –2 –4 –6 –8 –10 –5 –4 –3 –2 21 3 4 5 x f Figure 9 Solution Observe the graph of f. The graph attains a local maximum at x = 1 because it is the highest point in an open interval around x = 1. The local maximum is the y-coordinate at x = 1, which is 2. The graph attains a local minimum at... |
locally) and locating the highest and lowest points on the graph for the entire domain. The y-coordinates (output) at the highest and lowest points are called the absolute maximum and absolute minimum, respectively. To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the g... |
GEBRAIC fifi bh 5. f x=x 6. hx=−x − 8. g x=x −− 11. __ − k t=t+ t 9. y= x 7. 10. qx=x− + t − t t + pt= − 12. f x=x −b 13. g x=x − 14. px=x++h 15. kx=x−+h 16. f x=x +xx+h 17. gx=x−xx+h 18. at= +h t+ 19. bx= +h x+ 20. jx=x +h 21. rt=t +h 22. =+ 23. =+ + 26. f(x=x−xx, x+h 29. = + 24. 27. = = , + + 25. 28. = = + + +... |
__.b x fb __ − REAL-WORLD APPLICATIONS 54. 56. dt=ttdt t=t= 55. o fi 57. ThFigure 19 Time (days) Figure 19 t t=t= Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 3.4 COMPOSITION OF FUNCTIONS 209 LEARNING OBJECTIVES In this section, you will: • Combine functions using algebraic op... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.