Rational Functions

10 Key excerpts on "Rational Functions"

• 

  No longer available |Learn more

  Introductory Experimental Mathematics & Key Mathematical Concepts

  • (Author)
  • 2014(Publication Date)
  • Orange Apple

    (Publisher)

  Quotients of polynomials are called rational expressions, and functions that evaluate rational expressions are called Rational Functions. Rational Functions are formal quotients of polynomials (they are formed from polynomials just as rational numbers are formed from integers, writing a fraction of two of them; fractions related by the canceling of common factors are identified with each other). The Rational Functions contain the Laurent polynomials, but do not limit denominators to be powers of a variable. A rational function produces rational output for any rational input; this is not true other functions such as trigonometric functions, logarithms and exponential functions. Formal power series are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree. Unlike polynomials they cannot in general be explicitly and fully written down (just like real numbers cannot), but the rules for manipulating their terms are the same as for polynomials. ________________________ WORLD TECHNOLOGIES ________________________ Rational function In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Definitions Rational function of degree 2: In the case of one variable, , a function is called a rational function if and only if it can be written in the form where P(x) and Q(x) are polynomial functions of x . ________________________ WORLD TECHNOLOGIES ________________________ Examples Rational function of degree 3: The rational function is not defined at . The rational function is defined for all real numbers, but not for all complex numbers, since if x were the square root of − 1 (i.e. the imaginary unit) or its negative, then formal evaluation would lead to division by zero: , which is undefined. The rational function , as x approaches infinity, is asymptotic to . A constant function such as f ( x ) = π is a rational function since constants are polynomials.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Precalculus

  • Cynthia Y. Young(Author)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Rational Function DEFINITION The domain of any polynomial function is the set of all real numbers. When we divide two polynomial functions, the result is a rational function, and we must exclude any values of x that make the denominator equal to zero. ▼ A N S W E R The domain is the set of all real numbers such that x 2 21 or x 2 4. Interval notation: 12q , 212 ∪ 1 21, 42 ∪ 1 4, q2 It is important to note that there are not always restrictions on the domain. For example, if the denominator is never equal to zero, the domain is the set of all real numbers. EXAMPLE 1 Finding the Domain of a Rational Function Find the domain of the rational function ƒ 1 x 2 5 x 1 1 x 2 2 x 2 6 . Express the domain in interval notation. Solution: Set the denominator equal to zero. x 2 2 x 2 6 5 0 Factor. 1 x 1 2 21 x 2 3 2 5 0 Solve for x. x 5 22 or x 5 3 Eliminate these values from the domain. x 2 22 or x 2 3 State the domain in interval notation. 1 2q, 22 2 ∪1 22, 3 2 ∪1 3, q 2 YOUR TURN Find the domain of the rational function ƒ 1 x 2 5 x 2 2 x 2 2 3x 2 4 . Express the domain in interval notation. ▼ 248 CHAPTER 2 Polynomial and Rational Functions It is important to note that ƒ 1 x 2 5 x 2 2 4 x 1 2 , where x 2 22, and g 1 x 2 5 x 2 2 are not the same function. Even though ƒ 1 x 2 can be written in the factored form ƒ 1 x 2 5 1 x 2 2 21 x 1 2 2 x 1 2 5 x 2 2, its domain is different. The domain of g 1 x 2 is the set of all real numbers, whereas the domain of ƒ 1 x 2 is the set of all real numbers such that x 2 22. If we were to plot ƒ 1 x 2 and g 1 x 2 , they would both look like the line y 5 x 2 2. However, ƒ 1 x 2 would have a hole, or discontinuity, at the point x 5 22. 2.6.2 Vertical, Horizontal, and Slant Asymptotes If a function is not defined at a point, then it is still useful to know how the function behaves near that point. Let’s start with a simple rational function, the reciprocal function ƒ 1 x 2 5 1 x . This function is defined everywhere except at x 5 0.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Precalculus

  • Cynthia Y. Young(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  • Understand that the graph of a rational function can have either a horizontal asymptote or a slant asymptote but not both. • Any real number excluded from the domain of a rational function corresponds to either a vertical asymptote or a hole in the graph. 2.6.1 Domain of Rational Functions 2.6.1 Skill Find the domain of a rational function. 2.6.1 Conceptual Understand that the domain of a rational function is the set of all real numbers except those that correspond to the denominator being equal to zero. So far in this chapter we have discussed polynomial functions. We now turn our attention to Rational Functions, which are ratios of polynomial functions. Ratios of integers are called rational numbers. Similarly, ratios of polynomial functions are called Rational Functions. Rational Function A function f (x) is a rational function if f (x) = n(x) ____ d(x) d(x) ≠ 0 where the numerator, n(x), and the denominator, d(x), are polynomial functions. The domain of f (x) is the set of all real numbers x such that d(x) ≠ 0. Note: If d(x) is a constant, then f (x) is a polynomial function. The domain of any polynomial function is the set of all real numbers. When we divide two polynomial functions, the result is a rational function, and we must exclude any values of x that make the denominator equal to zero. Preview to Calculus In Exercises 87–92, refer to the following: In calculus we study the integration of Rational Functions by partial fractions. a. Factor each polynomial into linear factors. Use complex numbers when necessary. b. Factor each polynomial using only real numbers. 87. f (x) = x 3 + x 2 + x + 1 88. f (x) = x 3 − 6x 2 + 21x − 26 89. f (x) = x 4 + 5x 2 + 4 90. f (x) = x 4 − 2x 2 − 7x 2 + 18x − 18 91. f (x) = x 5 + 10x 4 + 50x 3 + 250x 2 + 625x 92. f (x) = x 5 − 7x 3 + 4x 2 − 20x

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  College Algebra

  Building Skills and Modeling Situations

  • Charles P. McKeague, Katherine Yoshiwara, Denny Burzynski(Authors)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  For example: The domain for f ( x) = x − 4 _____ x − 2 , is {x| x ≠ 2} A rational function is any function that can be written in the form f (x) = P(x) ____ Q(x) where P(x) and Q(x) are polynomials and Q(x) ≠ 0. Rational Function DEFINITION EXAMPLE 1 Graphing Rational Functions VIDEO EXAMPLES SECTION 5.5 382 Chapter 5 Polynomials and Rational Functions The domain for g( x) = x 2 + 5 _____ x + 1 , is {x| x ≠ −1} The domain for h( x) = x _____ x 2 − 9 , is {x| x ≠ −3, x ≠ 3} Notice that, for these functions, f (2), g(−1), h(−3), and h (3) are all undefined, and that is why the domains are written as shown. Graph the equation y = x 2 − 9 _____ x − 3 . How is this graph differ- ent from the graph of y = x + 3? SOLUTION We know from the discussion on factoring and reducing to lowest terms that y = x 2 − 9 _____ x − 3 = (x + 3)(x − 3) ___________ x − 3 = x + 3 This relationship is true for all x except x = 3 because the rational expressions with x − 3 in the denominator are undefined when x is 3. However, for all other values of x, the expressions x 2 − 9 _____ x − 3 and x + 3 are equal. Therefore, the graphs of y = x 2 − 9 _____ x − 3 and y = x + 3 will be the same except when x is 3. In the first equation, there is no value of y to correspond to x = 3. In the second equation, y = x + 3 so y is 6 when x is 3. Now you can see the difference in the graphs of the two equations. To show that there is no y value for x = 3 in the graph on the left in Figure 1, we draw an open circle at that point on the line. We say that there is a hole at that point. EXAMPLE 2 FIGURE 1 –10 –8 –6 –2 2 4 6 8 10 –10 –8 –6 –4 –2 4 6 8 10 x y y H11005 x 2 H11002 9 x H11002 3 y H11005 x H11001 3 y H11005 x 2 H11002 9 x H11002 3 –10 –8 –6 –2 2 4 6 8 10 –10 –8 –6 –4 –2 4 6 8 10 x y y H11005 x H11001 3 5.5 Graphing Rational Functions 383 Notice that the two graphs shown in Figure 1 are both graphs of functions.

  Sign up to read

  Learn more about book
• 

  No longer available |Learn more

  Precalculus: Mathematics for Calculus, International Metric Edition

  • James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  We assume that P1 x 2 and Q1 x 2 have no factor in com- mon. Even though Rational Functions are constructed from polynomials, their graphs look quite different from the graphs of polynomial functions. ■ Rational Functions and Asymptotes The domain of a rational function consists of all real numbers x except those for which the denominator is zero. When graphing a rational function, we must pay special atten- tion to the behavior of the graph near those x-values. We begin by graphing a very simple rational function. EXAMPLE 1 ■ A Simple Rational Function Graph the rational function f 1 x 2  1/ x , and state the domain and range. SOLUTION The function f is not defined for x  0. The following tables show that when x is close to zero, the value of 0 f 1 x 20 is large, and the closer x gets to zero, the larger 0 f 1 x 20 gets. Approaching 0  Approaching  Approaching 0  Approaching  x f x x c 0.1 10 0.01 100 0.00001 100,000 x f x x c 0.1 10 0.01 100 0.00001 100,000 We describe this behavior in words and in symbols as follows. The first table shows that as x approaches 0 from the left, the values of y  f 1 x 2 decrease without bound. In symbols, f 1 x 2 S  as x S 0  “y approaches negative infinity as x approaches 0 from the left” Domains of rational expressions are discussed in Section 1.4. For positive real numbers, 1 BIG NUMBER  small number 1 small number  BIG NUMBER DISCOVERY PROJECT Managing Traffic A highway engineer wants to determine the optimal safe driving speed for a road. The higher the speed limit, the more cars the road can accommodate, but safety requires a greater following distance at higher speeds. In this project we find a rational function that models the carrying capacity of a road at a given traffic speed.The model can be used to determine the speed limit at which the road has its maximum carrying capacity. You can find the project at www.stewartmath.com. © silver-john/Shutterstock.com Copyright 2017 Cengage Learning.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Precalculus

  Building Concepts and Connections 2E

  • Revathi Narasimhan(Author)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Before examining Rational Functions in general, we look at some specific examples. Analyzing a Simple Rational Function Let f ( x ) = 1 ______ x − 1 . a. What is the domain of f ? b. Make a table of values of x and f ( x ). Include values of x that are near 1 as well as larger values of x . c. Graph f by hand. d. Comment on the behavior of the graph. Solution a. The domain of f is the set of all values of x such that the denominator, x − 1 , is not equal to zero. This is true for all x ≠ 1 . In interval notation, the domain is ( −∞ , 1) ∪ (1, ∞ ). b. Table 1 is a table of values of x and f ( x ). Note that it contains some values of x that are close to 1 as well as some larger values of x . c. Graphing the data in Table 1 gives us Figure 1. Note that there is no value for f ( x ) at x = 1. d. Examining the graph of f ( x ) = 1 ____ x − 1 , we see that as the absolute value of x gets large, the value of the function approaches zero, even though it never actually reaches zero. Also, as the value of x approaches 1, the absolute value of the function gets large. Check It Out 1 What is the domain for f ( x ) = 1 _ x ? Sketch a graph of f . Just in Time Review rational expressions in Section 1.4. Example 1 VIDEO EXAMPLES SECTION 3.7 x f ( x ) = 1 _____ x − 1 − 100 − 0.009901 − 10 − 0.090909 0 − 1 0.5 − 2 0.9 − 10 1 Undefined 1.1 10 1.5 2 2 1 10 0.111111 100 0.010101 Table 1 2 4 6 8 y f ( x ) = –4 –2 –1 –6 –8 4 x 3 1 2 –2 –4 –3 x – 1 1 Figure 1 Objectives ■ Define a rational function. ■ Examine the end behavior of a rational function. ■ Find vertical asymptotes and intercepts. ■ Find horizontal asymptotes. ■ Sketch a complete graph of a rational function. Rational Functions 284 Chapter 3 Polynomial and Rational Functions When considering Rational Functions, we often use a pair of facts about the relationship between numbers that are large in absolute value and their reciprocals.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  College Algebra

  • James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  11. (a) What is a rational function? (b) What does it mean to say that x  a is a vertical asymp-tote of y  f 1 x 2 ? (c) What does it mean to say that y  b is a horizontal asymptote of y  f 1 x 2 ? 12. (a) How do you find vertical asymptotes of Rational Functions? (b) Let s be the rational function s 1 x 2  a n x n  a n  1 x n  1  . . .  a 1 x  a 0 b m x m  b m  1 x m  1  . . .  b 1 x  b 0 How do you find the horizontal asymptote of s ? (c) Find the vertical and horizontal asymptotes of f 1 x 2  5 x 2  3 x 2  4 13. (a) Under what circumstances does a rational function have a slant asymptote? (b) How do you determine the end behavior of a rational function? 14. (a) Explain how to solve a polynomial inequality. (b) What are the cut points of a rational function? Explain how to solve a rational inequality. (c) Solve the inequality x 2  9  8 x . ■ CONCEPT CHECK ANSWERS TO THE CONCEPT CHECK CAN BE FOUND AT THE BACK OF THE BOOK. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 356 CHAPTER 3 ■ Polynomial and Rational Functions 1–4 ■ Graphs of Quadratic Functions A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. 1. f 1 x 2  x 2  6 x  2 2. f 1 x 2  2 x 2  8 x  4 3. f 1 x 2  1  10 x  x 2 4. g 1 x 2   2 x 2  12 x 5–6 ■ Maximum and Minimum Values Find the maximum or minimum value of the quadratic function. 5. f 1 x 2   x 2  3 x  1 6. f 1 x 2  3 x 2  18 x  5 7. Height of a Stone A stone is thrown upward from the top of a building.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Precalculus

  Functions and Graphs, Enhanced Edition

  • Earl Swokowski, Jeffery Cole(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 3.5 Rational Functions 233 Selecting only to be graphed (turn off and ) and using a standard view-ing rectangle, we obtain a graph that gives us virtually no indication of the true shape of f . Changing to a viewing rectangle of by gives us a hint that the vertical asymptotes are confined to the interval . Using a viewing rectangle of by and changing to dot mode (so as not to graph the function across the vertical asymptotes) leads us to the sketch in Figure 17. Since the degree of the numerator, 2, is less than the degree of the denominator, 3, we know that the horizontal asymptote is the x -axis. The zeros of the numerator, 0 and 1, are the only x -intercepts. To determine the equations of the vertical asymptotes, we will abandon the graph of and examine the graph of —looking for its zeros. Graphing with the same viewing rectangle, but using connected mode, gives us Figure 18. By the theorem on rational zeros of a polynomial, we know that the pos-sible rational roots of are From the graph, we see that the only reasonable choice for the zero in the interval is . The number 2 appears to be a zero, and using a zero or root feature indicates that is also a good candidate for a zero. We can prove that , , and 2 are zeros of by using synthetic division. Thus, equations of the vertical asymptotes are ■ Graphs of Rational Functions may become increasingly complicated as the degrees of the polynomials in the numerator and denominator increase. Techniques developed in calculus are very helpful in achieving a more thor-ough treatment of such graphs. Formulas that represent physical quantities may determine rational func-tions.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Algebra and Trigonometry

  • Cynthia Y. Young(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Complex zeros Complex conjugate pairs: • If a + bi is a zero of P(x), then a − bi is also a zero. Factoring polynomials The polynomial can be written as a product of linear factors, not necessarily distinct. 4.6 Rational Functions f (x) = n(x) ____ d(x) d(x) ≠ 0 Domain of Rational Functions Domain: All real numbers except x-values that make the denominator equal to zero; that is, d(x) = 0. A rational function f (x) = n(x) ____ d(x) is said to be in lowest terms if n(x) and d(x) have no common factors. Horizontal, vertical, and slant asymptotes A rational function that has a common factor x − a in both the numerator and denominator has a hole at x = a in its graph if the multiplicity of a in the numerator is greater than or equal to the multiplicity of a in the denominator. Vertical Asymptotes A rational function in lowest terms has a vertical asymptote corresponding to any x-values that make the denominator equal to zero. Horizontal Asymptotes • y = 0 if degree of n(x) < degree of d(x). • No horizontal asymptote if degree of n(x) > degree of d(x). • y = Leading coefficient of n(x) ______________________ Leading coefficient of d(x) if degree of n(x) = degree of d(x). Slant Asymptotes If degree of n(x) − degree of d(x) = 1. Divide n(x) by d(x) and the quotient determines the slant asymptote; that is, y = quotient. Graphing Rational Functions 1. Find the domain of the function. 2. Find the intercept(s). 3. Find any holes. 4. Find any asymptotes. 5. Find additional points on the graph. 6. Sketch the graph: Draw the asymptotes and label the intercepts and points and connect with a smooth curve. Chapter 4 Review Exercises 417 4.1 Quadratic Functions Match the quadratic function with its graph. 1. f (x) = −2(x + 6) 2 + 3 2. f (x) = 1 _ 4 (x − 4) 2 + 2 3. f (x) = x 2 + x − 6 4. f (x) = −3x 2 − 10x + 8 a. b. –10 10 –10 10 x y –10 10 –10 10 x y c. d. –10 10 –10 10 x y –10 10 x y –4 16 Graph the quadratic function given in standard form.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Precalculus

  Functions and Graphs

  • Earl Swokowski, Jeffery Cole(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 3.5 Rational Functions 231 as x → 2 1 because there are no x -intercepts in R 3 . This implies that as x → ` , the graph approaches the horizontal asymptote from above, as in Figure 13(b). The graph of f is sketched in Figure 13(c). ■ In the remaining solutions we will not formally write down each guideline. EXAMPLE 8 Sketching the graph of a rational function Sketch the graph of f if f s x d 5 2 x 4 x 4 1 1 . Solution Note that since f s 2 x d 5 f s x d , the function is even, and hence the graph is symmetric with respect to the y -axis. The graph intersects the x -axis at s 0, 0 d . Since the denominator of f s x d has no real zero, the graph has no vertical asymptote. The numerator and denominator of f s x d have the same degree. Since the leading coefficients are 2 and 1, respectively, the line y 5 2 1 5 2 is the hori-zontal asymptote. The graph does not cross the horizontal asymptote y 5 2 , since the equation f s x d 5 2 has no real solution. Plotting the points s 1, 1 d and s 2, 32 17 d and making use of symmetry leads to the sketch in Figure 14. ■ An oblique (or slant) asymptote for a graph is a line y 5 ax 1 b , with a ± 0 , such that the graph approaches this line as x → ` or as x → 2` . (If the graph is a line, we consider it to be its own asymptote.) If the rational function f s x d 5 g s x dy h s x d for polynomials g s x d and h s x d and if the degree of g s x d is one greater than the degree of h s x d , then the graph of f has an oblique asymptote.

  Sign up to read

  Learn more about book