Concepts Examples Graph ƒ1x2 = x2 - 1 1x + 321x - 22 . x y 0 3 –3 1 –1 –5 1 5 x = –3 x = 2 y = 1 (5, 1) f(x) = (x + 3)(x – 2) x2 – 1 Graph ƒ1x2 = x2 - 1 x + 1 . ƒ1x2 = x2 - 1 x + 1 ƒ1x2 = 1x + 121x - 12 x + 1 ƒ1x2 = x - 1, x ≠-1 The graph is that of y = x - 1, with a hole at 1-1, -22. Graphing a Rational Function ƒ1x2 = p1 x2 q1 x2 To graph a rational function in lowest terms, follow these steps. Step 1 Find any vertical asymptotes. Step 2 Find any horizontal or oblique asymptotes. Step 3 If q102 ≠0, plot the y-intercept by evaluating ƒ102. Step 4 Plot the x-intercepts, if any, by solving ƒ1x2 = 0. Step 5 Determine whether the graph will intersect its nonvertical asymptote y = b or y = mx + b by solving ƒ1x2 = b or ƒ1x2 = mx + b. Step 6 Plot selected points, as necessary. Choose an x-value in each domain interval determined by the vertical asymptotes and x-intercepts. Step 7 Complete the sketch. Point of Discontinuity A rational function that is not in lowest terms often has a hole, or point of discontinuity, in its graph. Solve each inequality. The solution set of the inequality 2x3 + 5x2 - x - 6 60 includes the x-values for which the graph of ƒ1x2 lies below the x-axis. Thus, the solution set is 1-∞, -22 ´ a3 2 , 1b. The solution set of the inequality 1 - x x + 4 Ú 0 includes the x-values for which the graph of ƒ1x2 lies on or above the x-axis. Thus the solution set is 1-4, 14. x y 0 2 1 –1 –2 1 2 f(x) = x + 1 x2 – 1 6 –2 0 1 2 –1 –6 f(x) = 2x3 + 5x2 – x – 6 = (x – 1)(2x + 3)(x + 2) x y x y 1 3 –4 –1 0 x = –4 y = 21 f(x) = x + 4 1 – x 424 CHAPTER 3 Polynomial and Rational Functions 3.5 Rational Functions: Graphs, Applications, and Models 3.6 Polynomial and Rational Inequalities Solving a Polynomial Equation or Inequality Using a Graph Step 1 Rewrite the equation or inequality, if necessary, so that an expression is on one side with 0 on the other side. Step 2 Set the expression of the equation or inequality equal to ƒ1x2 and graph the related function. Step 3 Use the graph of ƒ1x2 to determine solutions as follows. • The real solutions of ƒ1x2 = 0 are the x-values of the x-intercepts of the graph. These are the zeros of ƒ1x2. • The real solutions of ƒ1x2 60 are the x-values for which the graph lies below the x-axis. • The real solutions of ƒ1x2 70 are the x-values for which the graph lies above the x-axis. Solving a Rational Equation or Inequality Using a Graph Follow the same steps as above. When solving a rational inequality, remember that a value that causes any denominator to equal 0 must be excluded from the solution set.
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