Sketch the graph of each rational function showing all the key features. Verify your graph by graphing the function onthe graphing calculator.3. f(x) = 3x − 2x2 / x − 2

Answer:The domain of the function is the set of all real numbers except 2.When x ⇒ 2⁻, f(x) ⇒ +∞When x ⇒ 2⁺, f(x) ⇒ -∞x = 2 is a vertical asymptote.When x ⇒ +∞, f(x) ⇒ -∞When x ⇒ -∞, f(x) ⇒ +∞At x = 0 and x = 3/2 the function intersect the x-axis.Step-by-step explanation:Hi there!First, let´s write the fucntion:f(x) = (3x - 2x²) / (x - 2)Let´s find the value of x for which the denominator is zero. That value will not be included in the domain of the function:x - 2 = 0x = 2Then, the domain of the function is the set of all real numbers except 2.If the function tends to infinity (∞) when "x" tends to 2, then x = 2 is a vertical asymptote. So let´s evaluate the behavior of the function when we near to 2 from left (x tends to 2 from the left = x ⇒2⁻) and from the right (x tends to 2 from the right = x ⇒2⁺).When x ⇒ 2⁻f(x) ⇒ 3 · 2 - 2 · 2²/2 - 2 ⇒6 - 8 /0 ⇒ -2/0 (since x tends to 2 from the left, x<2, then, x - 2<0).Then, a number divided by a very small number (almost zero) gives infinty. In this case, both numbers are negative, therefore:When x ⇒ 2⁻, f(x) ⇒ +∞The same procedure should be done to evaluate the behavior of the function when x nears 2 from the right. In this case, x>2, then x-2>0. The numerator will be the same, -2. Therefore:When x ⇒ 2⁺, f(x) ⇒ -∞Then, x = 2 is a vertical asymptote.Now let´s find if the function tends to a certain value when it tends to infinty. In that case, that value will be a horizontal asymptote of f(x). So:When x ⇒ +∞f(x) ⇒ (3x - 2x²) / x   (x>>>2, then it can be considered that x - 2 ≈ x)Let´s apply distributive property:f(x) ⇒ 3x/x - 2x²/xf(x) ⇒ 3 - 2x ⇒ -2x ⇒ -∞Then, when x ⇒ +∞, f(x) ⇒ -∞, there is no asymptote.In the same way, when x ⇒ -∞f(x) ⇒ -2x ⇒ +∞Now, let´s find the zeros of the function:f(x) = (3x - 2x²) / (x - 2)0 = (3x - 2x²) / (x - 2)0 = 3x - 2x²0 = x(3- 2x)    (x = 0)0 = 3-2x-3 = -2x-3/-2 = xx = 3/2Then at x = 0 and x = 3/2 the function intersect the x-axis.Attached, you can find the graph of the function.