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For the horizontal asymptote, we have, y = = = 3. Since the x and y intercepts are already in the left region, we won't need to get any points there. That means, we'll just get a point in the right region. Say, x = 2. 2-1 f(2) = (2)+6 - 12 = 12 (2,12) Plotting these points on the graph, we get the following. The definition of rational equations, inequalities and functions are summarized below. Rational Equation Rational Inequality Rational Function Definition An equation involving An inequality involving A function of the form rational expressions. rational expressions. q(x) f(x) = 2 where p(x) and q (x) are polynomial functions and q(x) is a nonzero function (Le., q(x) = 0). Example 2_3 1 X 2x #5 f(x) = *' + 2x +3 x+1 18 A rational equation or inequality can be solved for all x values that satisfy the equation or inequality. Whereas we solve an equation or inequality, we do not "solve" functions. Rather, a function (and in particular, a rational function) expresses a relationship between two variables (such as x and y), and can be represented by a table of values or a graph. Activity 3. Determine whether the given is a rational function, rational equation, rational inequality, or none of these. 1. y = 5x3 - 2x + 1 1 = x3 x+1 2. 9 - 8 = 7 5. g(x) = 2x-4x+1 x2+3 3. Vx - 2 =4 6. 6x - - 20 x+3 Activity 4. Find the solutions for each rational equation below. Make sure to check for extraneous solutions. 1. 4. X-4x _14-9x x+1 x-2 x-2 -2 = 2 2 7+1 2x 5. 2x-1 - = 5 x+3 3. x2-10 -14-5x x-1 x-1 Activity 5. Find the solution set for each rational inequality below. Graph the solution set on a number line. 1. (x+3)(x-2) > 0 4. x-2 (x+2)(x-1) x2-3x-10 -