88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers: { a a and b are integers and b } b When we discuss algebraic fractions, we can define them in a similar fashion: Rational Expression ( algebraic fractions ) A Rational Expression R(x) is the ratio of two polynomials P(x) Q(x) the polynomial Q(x) in the denominator is not (Q(x) ). Some examples of rational expressions include: R(x) = x x+ 9, R(x) =, and R(x) = 9x 6xy +9y x 7y Objective #: Finding the domain of a rational expression such that To Find the Domain of a Rational Expression. Set the denominator of the expression equal to zero and solve.. The domain is all real numbers except those values find in part. Find the domain of the following: Ex. a R(x) = 9 x Ex. b f(t) = t t+7 p Ex. c w(p) = 9 Ex. d h(x) = x x+ p 7p+ x + a) Set the denominator equal to zero and solve: x = x = Now, exclude x = from the real numbers. Thus, the domain is { x x }. b) Set the denominator equal to zero and solve: t + 7 = t = 7 Now, exclude t = 7 from the real numbers. Thus, the domain is { t t 7}.
89 c) Set the denominator equal to zero and solve: p 7p + = (factor) (p )(p ) = (set each factor equal to and solve) p = or p = p = or p = Now, exclude those values. Thus, the domain is { p p, p }. d) Set the denominator equal to zero and solve: x + = (factor out ) (x + 6) = (x + 6 is prime) Set each factor equal to zero and solve: = or x + 6 = No Solution No real solution since x 6 Hence, there are no values to exclude. Thus, the domain is set of all real numbers. After finding the domain, when working with graphing rational expressions, it is important to be sure the expression is reduced to lowest terms. For instance, compare f(x) = x and g(x) = x. Since f(x) reduces to g(x), x+ one might assume that their domains are the same, when if fact, they are not. The domain of g is all real numbers, but the domain of f is { x x }. The graph of f will have a hole at x = : Graph of f - - - - - - - - - - Graph of g - - - - - - - - - -
9 Now, let's explore the graph of R(x) =. The domain is { x x }. The graph will not have any intercepts since the y-intercept is undefined and setting = and solving to obtain the x-intercepts yields the equation: x = which has no solution. Let's make a table of values and plot some points: x R(x) x R(x) 6 6 x 6 6 8 7 6 - - - - - - - - - - - - - Notice that as x grows to be a larger positive number (we say x increases without bound and symbolically, x ) and as x grows to be a larger negative number (we say x decreases without bound and symbolically, x ), the value of R(x) gets closer to zero (R(x) ). In other words, as x gets to be a large positive or negative number, the graph of R(x) approaches the horizontal line y =. Likewise, as x gets closer to zero 8 7 6 -
(x ), the function value increases without bound (R(x) ). Thus, as x gets close to zero, the graph of R(x) approaches the vertical line x =. This leads to the following definition: Definition: Let R be a function of x. ) x = c is a vertical asymptote of R if as x c, R(x) ±. The graph of the function R(x) can never intersect a vertical asymptote. ) y = L is a horizontal asymptote of R if as x or as x, the function values R(x) L. It is possible for the graph to intersect a horizontal asymptote. ) y = mx + b is an oblique (slant) asymptote of R if as x or as x, the function values R(x) mx + b. It is possible for the graph to intersect an oblique asymptote. This graph has a This graph has a This graph has a vertical horizontal asymptote vertical asymptote asymptote of x = and a of y = of x =. horizontal asymptote of y =. - - - - - - - - - - - - - 7-6 - - - - - This graph has a V.A. This graph has a This graph has a V.A. of of x = and an O. A. horizontal asymptote x = and an O.A. of of y = x +. of y =. y =.x. 9 8 7 6 - - - - 6 7 7 6 - - - - - - - - 6 - - 6 - - - - - - - - - - - 6 9
Sketch the graph of the following using transformations. Identify the domain, the range, and any asymptotes: Ex. g(x) = Ex. h(x) = x (x+) The graph of g is the graph h(x) = x + x of stretched by a factor x of, reflected across the graph of x-axis and shifted left and down. + x = x + = +. x x x Thus, the graph of h is the x shifted up unit. 9-8 -7-6 - - - - - - - - - - -6-7 - - - - The domain is { x x } The domain is { x x } and the range is (, ). and the range is (, ). V.A.: x = V.A.: x = H.A.: y = H.A.: y = Objective # & #: Finding Asymptotes of a rational function. Vertical Asymptotes: If a R(x) = p(x) is a rational function reduced to lowest terms, the x = c is a q(x) vertical asymptote of R if q(c) =.
Horizontal Asymptotes: Case: If the degree of the polynomial in the denominator is equal to the degree of the polynomial in the numerator, then y = a b is a horizontal asymptote where a and b are the leading coefficients of the polynomials in the numerator and denominator respectively. Case : If the degree of the polynomial in the denominator is greater than the degree of the polynomial in the numerator, then y = is a horizontal asymptote. Case : If the degree in the polynomial in the numerator is larger than the degree of the polynomial in the denominator, then there is no horizontal asymptote. Oblique (Slant) Asymptotes: If the degree of the polynomial in the numerator is one more than the degree of the polynomial in the denominator, then the oblique asymptote is y = mx + b where mx + b is obtained by finding quotient of the numerator and denominator and ignoring the remainder. Note: A function cannot have both a horizontal asymptote and an oblique asymptote. Find all the asymptotes for the following: Ex. k(x) = x 8x x + x 6 First, factor the numerator and denominator to get it in lowest terms: x 8x = x + x 6 x(x 8) (x )(x+) Nothing reduces. The denominator is zero when x =. and x = (V.A.). Since the degrees of the polynomials in the top & bottom are equal, then y = =. is a horizontal asymptotes. Thus, our asymptotes are: V.A.: x = and x =. H.A.: y =. O.A.: None 9
9 Ex. r(x) = x +6 x +9x First, factor the numerator and denominator to get it in lowest terms: x +6 = x +6 x +9x x(x +9) Nothing reduces. The denominator is zero when x = (V.A.). Since the degree of the polynomial in the denominator is higher then the degree in the numerator, then y = is a horizontal asymptotes. Thus, our asymptotes are: V.A.: x = H.A.: y = O.A.: None Ex. 6 t(x) = x x x First, factor the numerator and denominator to get it in lowest terms: x x x = x (x ) x = x (x ) (x )(x+ ) Nothing reduces. The denominator is zero when x = & x = (V.A.). Since the degree of the polynomial in the numerator is one degree higher than the degree in the denominator, then there is no horizontal asymptotes, but there is an oblique asymptote. Divide the polynomials to find it: x x x x x + x x + x + x x Ignore the remainder. Thus, our asymptotes are: V.A.: x = and x = H.A.: None O.A.: y = x
9 Ex. 7 p(x) = 6x +x +x Ex. 8 x +x + x + First, factor the numerator and denominator to get it in lowest terms: 6x +x +x = 6x (x +x+ ) = 6x (x+)(x+) x +x + x + x (x+)+(x +) (x+)(x +) = 6x (x+) (x +) In the denominator, x +, then there are no vertical asymptotes. Since the degree of the polynomial in the numerator is one degree higher than the degree in the denominator, then there is no horizontal asymptotes, but there is an oblique asymptote. Divide the polynomials to find it: 6x + x + x + x + 6x + x + x 6x x 6x x x + 8x x x x x 6x x (Ignore) Thus, our asymptotes are: V.A.: None H.A.: None O.A.: y = 6x + r(x) = x 7 x 8 First, factor the numerator and denominator to get it in lowest terms: x 7 = (x )(x +x+9) x 8 (x 9)(x +9) = (x )(x +x+9) (x )(x+)(x +9) = (x +x+9). The denominator is zero when x = (V.A.). (x+)(x +9) Since the degree of the polynomial in the denominator is higher then the degree in the numerator, then y = is a horizontal asymptotes. Thus, our asymptotes are: V.A.: x = H.A.: y = O.A.: None
96 Ex. 9 k(x) = 6x + x x x x x First, factor the numerator and denominator to get it in lowest terms: 6x + x x = x(6x + x ) = x(x+)(x ) x x x x(x x ) x(x+)(x ) = (x +)(x ) (x +)(x ) The denominator is zero when x = / and x = (V.A.). Since the degrees of the polynomials in the top & bottom are equal, then y = 6 = is a horizontal asymptotes. Thus, our asymptotes are: V.A.: x = / and x = H.A.: y = O.A.: None Ex. p(x) = x 6 x x First, factor the numerator and denominator to get it in lowest terms: x 6 x x = (x )(x +x+6) x(x ) = (x +x+6) x The denominator is zero when x = (V.A.). Since the degree of the polynomial in the numerator is one degree higher than the degree in the denominator, then there is no horizontal asymptotes, but there is an oblique asymptote. Divide the polynomials to find it: x x + x x 6 x + x x 6 x + 6x 6x 6 (Ignore) Thus, our asymptotes are: V.A.: x = H.A.: None O.A.: y = x. Objective #: Multiplicity of Vertical Asymptotes. We use the multiplicity of zeros to determine if the graph of a polynomial function touched (even multiplicity) or crossed (odd multiplicity) at a
97 particular x-intercept. We can also look at the multiplicity of vertical asymptotes to determine how the graph will behave on either side of the vertical asymptote. Vertical Asymptotes and Multiplicity Let x = a be a vertical asymptote of a function R. ) If the multiplicity of the vertical asymptote is odd, then R will approach on one side of the vertical asymptote and on the other side of the vertical asymptote. OR ) If the multiplicity of the vertical asymptote is even, then R will approach either on both sides of the vertical asymptote or on both sides of the vertical asymptote