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Math 1314 Rational Functions Rational Functions A rational function is the algebraic equivalent of a rational number. Recall that a rational number is one that can be expressed as a ratio of integers: p/q. Examples: 2/3, -23 ( = -23/1), 0.005 ( = 5/1000) A rational function, by analogy, is a function that can be expressed as a ratio of polynomials: 1 3x22 7 x 2 x 1 x Examples: f( x ) , f( x ) , g( x ) , g( x ) x x22 x14 x x Domains and Ranges Notice that the domains of most rational functions must be restricted to values of x that will not make the denominator of the function equal to zero. In order to find the values of x to exclude from the domain, set the denominator equal to zero and solve for x. x Example: f (x) Do this: x2 - 4 0 x 2 4 (x - 2)(x + 2) 0 x - 2 0 x + 2 0 x 2 x -2 Therefore, the domain of f(x) is all real numbers, except for x = 2 and x = -2. The range of a rational function is sometimes easier to find by first finding the inverse of the function and determining its domain (remember that the range of a function is equal to the domain of its inverse). If this doesn’t work, the best strategy is to graph the rational function. To do that, you have to locate all asymptotes, as described below. Asymptotes In general, an asymptote is a line (or a curve) that the graph of a function gets close to but does not touch. There are three main types of asymptotes: 1. Vertical Asymptotes The vertical line x = c is a vertical asymptote of the graph of f(x), if f(x) gets infinitely large or infinitely small as x gets close to c. The graph of f(x) can never cross or touch the asymptote, x = c. i.e. as x c, f(x) or f(x) - . Finding Vertical Asymptote(s) A rational function reduced to lowest terms (all factors common to both numerator and denominator cancelled out) will have a vertical asymptote at every value of x that would make the denominator equal zero. One function may have many vertical asymptotes. Another way of looking at vertical asymptotes is that they are the restrictions of the domain of a reduced rational function. Refer to the domain example above, x x = 2 and x = -2 would be the vertical asymptotes of f (x) . x 2 4 Special Case: Holes The factors that are cancelled when a rational function is reduced represent holes in the graph of f(x). xx2 32(x 1) (x 2) x 2 Example: fx() xx2 43(x 1) (x 3) x 3 Instead of having two vertical asymptotes at x = 1 and x = 3, this rational function has one hole at x = 1 and one vertical asymptote at x = 3. 2. Horizontal Asymptotes The line y = b is a horizontal asymptote for the graph of f(x), if f(x) gets close b as x gets really large or really small. i.e. as x , f(x) b Note that f(x) can approach its horizontal asymptote from either above or below, and the graph of f(x) may actually cross or intersect its horizontal asymptote at some central point. Finding Horizontal Asymptote A given rational function will either have only one horizontal asymptote or no horizontal asymptote. Case 1: If the degree of the numerator of f(x) is less than the degree of the denominator, i.e. f(x) is a proper rational function, the x-axis (y = 0) will be the horizontal asymptote. Case 2: If the degree of the numerator of f(x) equals the degree of the denominator, there will be a horizontal asymptote at the line y = b, where the constant b is determined by dividing the leading coefficient in the numerator by the leading coefficient of the denominator. LSC-Montgomery Learning Center: Rational Functions Page 2 Last Updated April 13, 2011 3x 2 7x 4 Examples: (a) f (x) Horizontal asymptote at y = 0 2x 4 5 x 2 1 (b) f (x) Horizontal asymptote at y = 2x 7 2 3. Oblique Asymptotes (a.k.a. diagonal or slant) The line y = mx + b is an oblique asymptote for the graph of f(x), if f(x) gets close to mx + b as x gets really large or really small. i.e. as x , f(x) mx + b Note that f(x) can approach its oblique asymptote from either above or below, and the graph of f(x) may cross or intersect its oblique asymptote at a (usually) central point. Finding Oblique Asymptote A given rational function will either have only one oblique asymptote or no oblique asymptote. If a rational function has a horizontal asymptote, it will not have an oblique asymptote. Oblique asymptotes only occur when the numerator of f(x) has a degree that is one higher than the degree of the denominator. When you have this situation, simply divide the numerator by the denominator, using polynomial long division or synthetic division. The quotient (set equal to y) will be the oblique asymptote. Note that the remainder is ignored. 8xx2 3 1 Example: fx() x 2 Using polynomial long division: 8x 13 Quotient x2 8 x2 3 x 1 8xx2 16 13x 1 13x 26 27 LSC-Montgomery Learning Center: Rational Functions Page 3 Last Updated April 13, 2011 Alternatively, using synthetic division: 2 8 -3 1 16 26 Quotient 8 13 |27 So, f(x) has an oblique asymptote at y = 8x + 13 Some Steps to Follow in Graphing Rational Functions 1. Find all asymptotes and plot them. 2. Find x-intercept(s) by setting the numerator equal to 0 and solving for x. 3. Find y-intercept by evaluating f(0). 4. If there is a horizontal asymptote y = b, determine whether the graph intersects this asymptote by solving f(x) = b. If x can be solved, the intersection is located at this x- value. If there is no solution for x, f(x) does not intersect the horizontal asymptote. 5. Use the vertical asymptotes and x-intercepts to divide the x-axis into intervals. Then determine whether f(x) is positive or negative on each of these intervals by using test points. 6. Determine the behavior of f(x) as x gets really large or really small (x ). If there is a horizontal or oblique asymptote, f(x) will approach the asymptote. Use the results in step 5 to determine whether the graph approaches from above or below. 7. If there is no horizontal or oblique asymptote, the highest degree term “takes over.” If you form the ratio of the numerator’s leading term to the denominator’s leading term and reduce this ratio to its lowest terms, f(x) will look very much like this function when x is very small or very large (note that this ratio will have degree 2 or greater when there is no horizontal or oblique asymptote). 3x 4 4 Example: f (x) x 1 3x4 Form ratio 3x3 x So, f(x) will be similar to the function y = 3x3 when x 8. Determine the behavior of f(x) as x approaches each vertical asymptote, both from the left side and from the right side. The function has to either get really big ( ) or really small ( ), and the results from step 5 will tell you which. LSC-Montgomery Learning Center: Rational Functions Page 4 Last Updated April 13, 2011 9. You may be able to make use of y-axis and origin symmetries -- check to see. 10. Connect all pieces and points with a smooth curve, plotting any additional test points you need for clarity. An Example of Graphing a Rational Function x Example: f (x) f(x) x 2 4 x=- 2 x=2 Asymptotes 1. Vertical: x = 2, x = -2 2. Horizontal: y = 0 3. Oblique: none Intercepts x-intercept = y-intercept = (0,0) Note that this is also the point where f(x) intersects its horizontal asymptote. Interval Interval Test Sign Behavior of f(x) Notation x-value of f(x) at asymptotes - < x < -2 (- , -2) -3 f(-3) = -3/5 as x , f(x) 0 from below = negative as x -2 from the left, f(x) - -2 < x < 0 (-2, 0) -1 f(-1) = 1/3 as x -2 from the right, f(x) = positive f(x) passes through the point (0,0) 0 < x < 2 (0, 2) 1 f(1) = -1/3 as x 2 from the left, f(x) - = negative 2 < x < (2, ) 3 f(3) = 3/5 as x 2 from the right, f(x) = positive as x , f(x) 0 from above LSC-Montgomery Learning Center: Rational Functions Page 5 Last Updated April 13, 2011 Determining Asymptotes of Rational Functions Reduce the rational function to its lowest terms When numerator and denominator have factor(s) in common there is a “hole” at that zero(s) Arrange both the numerator and denominator in descending order by degree axm bxn Vertical Asymptotes Set bxn 0 x x1 , x x2 , There may be one or more or none Other Asymptotes a) If m < n, then y = 0 (horizontal) a b) If m = n, then y (horizontal) b c) If m = n + 1, divide the rational function and the quotient represents an oblique asymptote in the form of y = mx + b There may be no vertical, horizontal or oblique asymptotes. A function cannot have both horizontal & oblique asymptotes.
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