1. For the rational function () = 3 1 + 1, determine the following key features, if they exist (no justification is required). [A - 7; 0.5 points each] a) Domain: ________________________________________________________________ b) Range: _________________________________________________________________ c) X-Intercept(s): ____________________________________________________________ d) Y-Intercept: ______________________________________________________________ e) Vertical asymptote equation(s): ______________________________________________ f) Horizontal asymptote equation(s): ____________________________________________ g) Behaviour(s) near the vertical asymptote(s): ____________________________________ h) Behaviour near the horizontal asymptote: ______________________________________ i) Interval(s) where the function is increasing: _____________________________________ j) Interval(s) where the function is decreasing: ____________________________________ k) Interval(s) where the function is positive: _______________________________________ l) Interval(s) where the function is negative: ______________________________________ m) Local maximum and/or minimum point(s): ______________________________________ n) Accurate and labeled sketch of the function (use the space below). 2 2. For the rational function () = 2 25 24 , determine the following key features, if they exist (no justification is required). [A - 7; 0.5 points each] a) Domain: ________________________________________________________________ b) Range: _________________________________________________________________ c) X-Intercept(s): ____________________________________________________________ d) Y-Intercept: ______________________________________________________________ e) Vertical asymptote equation(s): ______________________________________________ f) Horizontal asymptote equation(s): ____________________________________________ g) Behaviour(s) near the vertical asymptote(s): ____________________________________ h) Behaviour near the horizontal asymptote: ______________________________________ i) Interval(s) where the function is increasing: _____________________________________ j) Interval(s) where the function is decreasing: ____________________________________ k) Interval(s) where the function is positive: _______________________________________ l) Interval(s) where the function is negative: ______________________________________ m) Local maximum and/or minimum points: _______________________________________ n) Accurate and labeled sketch of the function (use the space below). 3 3. Determine a possible equation of each of the following rational functions given their descriptions. Write the final answers only using appropriate rational forms. [T/I - 8] a) A rational function which has a vertical asymptote with equation x = 6, a horizontal asymptote with equation y = 0, a y-intercept at the point (0, 0.5) and no x-intercepts. b) A rational function which has no vertical asymptotes, a horizontal asymptote with equation y = 0, no x-intercepts and a local maximum point at the y-intercept of (0, 10). c) A rational function which has an x-intercept at the point (0.75, 0), a y-intercept at the point (0, - 3), a vertical asymptote with equation x = - 0.5 and a horizontal asymptote with equation y = 2. d) A rational function which has two vertical asymptotes with equations x = 3 and x = 1, a horizontal asymptote with equation y = 0, does not have any x-intercepts and a y-intercept at the point (0, - 1). 4 4. Solve the following rational equation and inequality using algebraic methods. Show all work and calculations. State any restrictions on the variables. [T/I - 8] a) 3 1 + 2 = 5 + 2 2 9 b) 2 30 2 12