2.4Polynomial And Rational Functions Polynomial Functions

2.4Polynomial and Rational FunctionsPolynomial FunctionsGiven a linear function f (x) mx b, we can add a square term, andget a quadratic function g(x) ax2 f (x) ax2 mx b. We cancontinue adding terms of higher degrees, e.g. we can add a cube termand get h(x) cx3 g(x) cx3 ax2 mx b, and so on. f (x), g(x),and h(x) are all special cases of a polynomial function.Definition (Polynomial Function)A polynomial function is a function that can be written in theformf (x) an xn an 1 xn 1 . . . a1 x a0for n a nonnegative integer, called the degree of the polynomial.The coefficients an , an 1 , . . . , a1 , a0 are real numbers with an 6 0.Note that although an 6 0, the remaining coefficients an 1 , an 2 , . . . , a1 , a0can very well be 0.Domain of Polynomial FunctionThe domain of a polynomial function is R, the set of all realnumbers.The domain of f (x) xn is R regardless the value of n (any nonnegative integer), and so is the domain of g(x) axn , where a is somereal number. Clearly, if you add, say k, such functions with differentdegrees (n) the domain of the resulting function will still be R.

Ch 2. Functions and Graphs2.4 Polynomial and Rational FunctionsConsider a function f (x) (x 1)(x 2)(x 3). It could be rewrittenasf (x) (x 1)(x 2)(x 3) 2 2 (x 1)(x 2x 3x 6) (x 1)(x 5x 6) x3 5x2 6x x2 5x 6 x3 6x2 11x 6.So, f (x) is a polynomial function of degree 3.Question: How many y intercepts does f (x) have?Answer: Only one, y f (0) 6. Any function can have at mostone y intercept, otherwise it will not pass the vertical line test.y Intercept of a Polynomial FunctionIf f (x) an xn an 1 xn 1 . . . a1 x a0 is a polynomial function,it has exactly one y intercept y a0 .Question: How many x intercepts does f (x) have?Answer: f (x) has 3 intercepts. 0 (x 1)(x 2)(x 3) x 1or x 2 or x 3.x Intercept of a Polynomial FunctionA polynomial of degree n can have, at most, n linear factors.Therefore, the graph of a polynomial function of positive degree ncan intersect the x axis at most n times. The x intercepts off (x) an xn an 1 xn 1 . . . a1 x a0 could be found by solvingan xn an 1 xn 1 . . . a1 x a0 0.2

Ch 2. Functions and Graphs2.4 Polynomial and Rational Functionsy765f (x)43210 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 x 1 2 3 4 5 6 7Consider a function h(x) (x2 1)(x 2)(x 3).h(x) (x2 1)(x 2)(x 3) (x2 1)(x2 2x 3x 6) (x2 1)(x2 5x 6) x4 5x3 6x2 x2 5x 6 x4 5x3 7x2 5x 6.h(x) is a polynomial function of degree 4, but has just 2 x intercepts,because the equation 0 (x2 1)(x 2)(x 3) has just 2 roots (zeros),which are x 2 and x 3.3

Ch 2. Functions and Graphs2.4 Polynomial and Rational Functions20yh(x)161284 101234x 4Note that f (x) x3 6x2 11x 6 has degree 3, which is an oddnumber. It starts negative, ends positive, and crosses the x axis oddnumber of times.h(x) x4 5x3 7x2 5x 6 has degree 4, which is an even number.It starts positive, ends positive, and cross the x axis even number oftimes.Consider m(x) f (x) (x3 6x2 11x 6) x3 6x2 11x 6,and n(x) g(x) (x4 5x3 7x2 5x 6) x4 5x3 7x2 5x 6.y7y854m(x)n(x)3 110 4x 7 6 5 4 3 2 1 1 0 1 2 3 4 5 6 7 8 3 12 5 16 7 2041234 x

Ch 2. Functions and Graphs2.4 Polynomial and Rational FunctionsDefinition (Leading Coefficient)Given a polynomial function f (x) an xn an 1 xn 1 . . . a1 x a0 ,the coefficient an of the highest-degree term is called the leadingcoefficient of a polynomial function f (x).Graph of a Polynomial FunctionGiven a polynomial function f (x) an xn an 1 xn 1 . . . a1 x a0 :(a) if an 0 and n is odd, then the graph of f (x) starts negative, ends positive, and crosses the x axis odd number oftimes but at least once;(b) if an 0 and n is odd, then the graph of f (x) starts positive, ends negative, and crosses the x axis odd number oftimes but at least once;(c) if an 0 and n is even, then the graph of f (x) startspositive, ends positive, and crosses the x axis even numberof times or does not cross it at all;(d) if an 0 and n is even, then the graph of f (x) startsnegative, ends negative, and crosses the x axis evennumber of times or does not cross it at all.Note: (c) is a reflection in the x axis of (a), and (d) is areflection in the x axis of (b).Also note that a polynomial function always either increases or decreases without bound as x goes to either negative or positive infinity.5

Ch 2. Functions and Graphs2.4 Polynomial and Rational FunctionsContinuity and ”Smoothness” of Polynomial FunctionConsider f (x) 2 x x .f (x) has a discontinuous break at x 0.y4f (x)2 5 3 1135x 2 46

Ch 2. Functions and Graphs2.4 Polynomial and Rational FunctionsConsider g(x) x 2. g(x) is continuous, but not smooth due to asharp corner at (0, 2).y4g(x)2 5 3 113Consider h(x) 2x 1 .5x 2 4h(x) has a discontinuous break at x 1.y5h(x)31 5 3 1 1135x 3 5Graph of a Polynomial FunctionThe graph of a polynomial function is continuous, with no holesor breaks. That is, the graph can be drawn without removing apen from the paper. Also, the graph of a polynomial is ”smooth”,i.e. has no sharp corners.7

Ch 2. Functions and Graphs2.4 Polynomial and Rational FunctionsRational FunctionsJust as rational numbers are defined in terms of quotients of integers,rational functions are defined in terms of quotients of polynomials.Definition (Rational Function)A rational function is any function that can be written in theformn(x), d(x) 6 0f (x) d(x)where n(x) and n(x) are polynomials.For example,f (x) 1,xg(x) x 2,x2 x 6p(x) x4 5x3 7x2 ,h(x) q(x) 123,x13 8,x5r(x) 0are all rational functions.If n(x) and d(x) are polynomials, then they both have domain R.However,Domain of a Rational FunctionIf f (x) n(x)d(x) is a rational function, then its domain is the set ofall real numbers such that d(x) 6 0.8

Ch 2. Functions and GraphsExample 1Find the domain of f (x) 2.4 Polynomial and Rational Functionsx2 1x2 7x 109

Ch 2. Functions and Graphs2.4 Polynomial and Rational FunctionsVertical and Horizontal AsymptotesRecall that a polynomial function is always continuous and ”smooth”.It is also true that if x increases or or decreases without bound, thenfunction also increases or decreases without bound. However, thismay not be true for a rational function. Also, a rational functionmay not have a y intercept.Consider a rational function f (x) x 3x 2 . Its domain ( , 2] [2, ),or all real numbers except for x 0012.0012.012.052.12.252.51.5 31.5 21.75 31.75 21.9 31.9 21.95 31.95 21.99 31.99 21.999 31.999 21.9999 31.9999 21.99999 31.99999 2f (x) 1.5 0.5 1.25 0.25 1.1 0.1 1.05 0.05 1.01 0.01 1.001 0.001 1.0001 0.0001 1.00001 0.000012.00001 32.00001 22.0001 32.0001 22.001 32.001 22.01 32.01 22.05 32.05 22.1 32.1 22.25 32.25 22.5 32.5 2 10 0.999990.00001 0.99990.0001 0.9990.001 0.990.01 0.950.05 0.90.1 0.750.25 0.50.5 3 5 11 21 101 1001 10001 100001undefined 99999 9999 999 99 19 9 3 1

Ch 2. Functions and Graphs2.4 Polynomial and Rational 00100010000100000 100000 3 100000 2 10000 3 10000 2 1000 3 1000 2 1000 3 1000 2 10 3 10 2 5 3 5 20 30 21.5 31.5 22.5 32.5 23 33 25 35 210 310 2100 3100 21000 31000 210000 310000 2100000 3100000 2f (x) 100003 100002 10003 10002 1003 1002 1003 1002 13 12 8 7 3 2 1.5 0.5 0.5 0.5 01 23 78 9798997 998 99979998 9999799998 1.00001 1.0001 1.001 1.001 1.08333 1.14286 1.5 3 1 0 0.666667 0.875 0.989796 0.998998 0.9999 0.99999x 2y8642 8 6 4 2 0 2 4 6246The graph of f (x) gets closer tothe line x 2 as x gets closer to 2.Line x 2 is a vertical asymptote for f (x).y 1The graph of f (x) gets closer to8xthe line y 1 as x increases or decreases without bound. The liney 1 is a horizontal asymptotefor f (x). 811

Ch 2. Functions and Graphs2.4 Polynomial and Rational FunctionsDefinition (Vertical Asymptote)A vertical line x a is called a vertical asymptote for a functionf (x) if the graph of f (x) gets closer to the line x a as x getscloser to a.Note: the number of vertical asymptotes of a rational function f (x) at most equal to the degree of d(x).n(x)d(x)isDefinition (Horizontal Asymptote)A horizontal line y b is called a horizontal asymptote for afunction f (x) if the graph of f (x) gets closer to the line y b as xgets x increases or decreases without bound.Note: a rational function has at most one horizontal asymptote. Moreover, thegraph of a rational function approaches the horizontal asymptote (when one exists)both as x increases and decreases without bound.x 2x 2yy 5 3f (x) 1 1 2 3 4 5x2x 05432154321135x 588 4(x 2)(x 2) 3 1 1 2 3 4 5f (x) x 12131x2 1 xx5x

Ch 2. Functions and Graphs2.4 Polynomial and Rational FunctionsExample 2Given the rational function f (x) 3x 3,x2 9(a) Find the domain.(b) Find the x and y intercepts.(c) Find the equations of all vertical asymptotes.(d) If there is a horizontal asymptote, find its equation.(e) Using the information from parts (a)-(d) and additional points as necessary,sketch a graph of f for 10 x 10.13

Ch 2. Functions and Graphs2.4 Polynomial and Rational Functionsy10987654321x 10 9 8 7 6 5 4 3 2 1 1 0 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 1014

Ch 2. Functions and Graphs2.4 Polynomial and Rational FunctionsConsider the rational function g(x) x 3y3x2 3x 36x3 4x2 9x 36.x 310987654321x 10 9 8 7 6 5 4 3 2 1 1 0 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 1015

Ch 2. Functions and Graphs2.4 Polynomial and Rational FunctionsExample 3Find the vertical and horizontal asymptotes of the rational function3f (x) xx2 4x 5x .16

Ch 2. Functions and Graphs2.4 Polynomial and Rational Functions17

Ch 2. Functions and Graphs2.4 Polynomial and Rational FunctionsApplicationsExample 4 (Employee Training)A company that manufactures computers has established that, onthe average, a new employee can assemble N (t) components perday after t days of on-the-job training, as given byN (t) 25t 5,t 5t 0Sketch a graph of N , 0 t 100, including any vertical or horizontal asymptotes. What does N (t) approach as t increases withoutbound?18

Ch 2. Functions and Graphs2.4 Polynomial and Rational FunctionsN (t)t19

Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions Rational Functions Just as rational numbers are de ned in terms of quotients of integers, rational functions are de ned in terms of quotients of polynomials. De nition (Rational Function) A rational function is any function that can be written in the form f(x) n(x) d(x); d(x) 6 0