Rational_Function.pdf

Rational Function  A rational function, 𝑟(𝑥) is a function of the form 𝑝(𝑥) 𝑟 𝑥 = 𝑞(𝑥) where 𝑝(𝑥) and 𝑞(𝑥) are polynomial functions and 𝑞 𝑥 ≠ 0 Identify the domain and range of the following and graph them using your Geogebra app 1...

Rational Function  A rational function, 𝑟(𝑥) is a function of the form 𝑝(𝑥) 𝑟 𝑥 = 𝑞(𝑥) where 𝑝(𝑥) and 𝑞(𝑥) are polynomial functions and 𝑞 𝑥 ≠ 0 Identify the domain and range of the following and graph them using your Geogebra app 1 𝑥 𝑓 𝑥 = 𝐹 𝑥 = 𝑥 𝑥−1 1 2𝑥 𝑔 𝑥 = 𝐺 𝑥 = 𝑥+1 𝑥+3  1 𝑓 𝑥 = 𝑥 x cannot be zero because it will render the function as undefined. Thus the domain: 𝑥 = {𝑥|𝑥 ≠ 0} and whatever you input in x (of course except 0) you’ll end up with any real number except 0 x 1 2 -1 -3 39 -70 -1/2 1/3 1/18 … y 1 1/2 -1 -1/3 1/9 -1/70 -2 3 18 … Thus the range: 𝑦 = {𝑦|𝑦 ≠ 0}  1 The graph of 𝑓 𝑥 = 𝑥 If you have noticed, the curves don’t seem to touch the axes  1 𝑔 𝑥 = 𝑥+1 x cannot be -1 because it will make the denominator 0 and render the function as undefined. Thus the domain: 𝑥 = {𝑥|𝑥 ≠ −1} and whatever you input in x (of course except -1) you’ll end up with any real number except 0 x 1 2 3 -3 -1/2 -2/3 -70 1/3 1/4 … y 1/2 1/3 1/4 -1/2 2 3 -1/69 3/4 4/5 … Thus the range: 𝑦 = {𝑦|𝑦 ≠ 0}  1 The graph of 𝑔 𝑥 = 𝑥+1 If you have noticed, the curves don’t seem to touch the line x =-1 and the x-axis  𝑥 𝐹 𝑥 = 𝑥−1 x cannot be 1 because it will make the denominator 0 and render the function as undefined. Thus the domain: 𝑥 = {𝑥|𝑥 ≠ 1} and whatever you input in x (of course except 1) you’ll end up with any real number except 1 x 0 2 3 -3 -1/2 2/3 -50 79 -200 … y 0 2 3/2 3/4 -1/3 -2 50/51 79/78 200/201 … Thus the range: 𝑦 = {𝑦|𝑦 ≠ 1}  𝑥 The graph of 𝐹 𝑥 = 𝑥−1 If you have noticed, the curves don’t seem to touch the line x=1 and the line y=1  2𝑥 𝐺 𝑥 = 𝑥+3 x cannot be -3 because it will make the denominator 0 and render the function as undefined. Thus the domain: 𝑥 = {𝑥|𝑥 ≠ −3} and whatever you input in x (of course except 3) you’ll end up with any real number except 2 x 0 2 3 -4 -1/2 2/3 -50 79 -200 … y 0 4/5 1 8 -2/5 -2 50/51 79/78 200/201 … Thus the range: 𝑦 = {𝑦|𝑦 ≠ 2}  2𝑥 The graph of 𝐺 𝑥 = 𝑥+3 If you have noticed, the curves don’t seem to touch the line x=-3 and the line y=2 If you have noticed, the domain and range is somehow the basis for that imaginary line wherein the graph will not cross. That imaginary line is called asymptote.  Techniques in graphing 𝑝(𝑥) rational functions 𝑟 𝑥 = 𝑞(𝑥) if 𝑝(𝑥) and 𝑞(𝑥) are st 1 degree polynomials or 𝑝(𝑥) is a constant and 𝑞(𝑥) is 1st degree Example 1: 𝑥−2 Graph 𝑓 𝑥 = 𝑥−3 First, identify the domain and range of the function: domain: 𝑥 = {𝑥|𝑥 ≠ 3} range: 𝑦 = {𝑦|𝑦 ≠ 1} These will determine our asymptotes. The domain will determine the vertical asymptote, while the range for the horizontal asymptote. Thus, the vertical asymptote will be the line x = 3 the horizontal asymptote will be the line y = 1  𝒙−𝟐 Graph 𝒇 𝒙 = 𝒙−𝟑 vertical asymptote x=3 horizontal asymptote y=1 Next, identify some points on the graph. Better draft the table of values: 𝑥−2 𝑓 𝑥 = 𝑥−3 Tip: On deciding values for x, choose values to the left of the vertical asymptote and to the right as well. In this case, the vertical asymptote is x = 3 thus to the left are {1,2} and to the right are {4,5}. x 1 2 4 5 … y 1/2 0 2 3/2 … And plot these points on the x-y plane. Extend the points with curves approaching but not touching the asymptotes.  𝒙−𝟐 Graph 𝒇 𝒙 = 𝒙−𝟑 vertical asymptote x=3 horizontal asymptote y=1  x 1 2 4 5 … y 1/2 0 2 3/2 … After plotting these points on the x-y plane. Extend the points with curves approaching but not touching the asymptotes. Remember that these curves mirror each other diagonally.  𝒙−𝟐 Graph 𝒇 𝒙 = 𝒙−𝟑 vertical asymptote x=3 horizontal asymptote y=1 Example 2: 2𝑥−1 Graph 𝑔 𝑥 = 𝑥+3 First, identify the domain and range of the function: domain: 𝑥 = {𝑥|𝑥 ≠ −3} range: 𝑦 = {𝑦|𝑦 ≠ 2} These will determine our asymptotes. The domain will determine the vertical asymptote, while the range for the horizontal asymptote. Thus, the vertical asymptote will be the line x = -3 the horizontal asymptote will be the line y = 2  2𝑥−1 Graph 𝑔 𝑥 = 𝑥+3 vertical asymptote x = -3 horizontal asymptote y=2 Next, identify some points on the graph. Better draft the table of values: 2𝑥 − 1 𝑔 𝑥 = 𝑥+3 Tip: On deciding values for x, choose values to the left of the vertical asymptote and to the right as well. In this case, the vertical asymptote is x = -3 thus to the left are {-5,-4} and to the right are {-2,-1}. x -5 -4 -2 -1 … y 11/2 9 -5 -3/2 … And plot these points on the x-y plane. Extend the points with curves approaching but not touching the asymptotes.  2𝑥−1 Graph 𝑔 𝑥 = 𝑥+3 c vertical asymptote x = -3 horizontal asymptote y=2  x -5 -4 -2 -1 … y 11/2 9 -5 -3/2 … After plotting these points on the x-y plane. Extend the points with curves approaching but not touching the asymptotes. Remember that these curves mirror each other diagonally.  2𝑥−1 Graph 𝑔 𝑥 = 𝑥+3 c vertical asymptote x = -3 horizontal asymptote y=2 Graph the following 𝑥−2 2−3𝑥 𝑓 𝑥 = 𝐹 𝑥 = 𝑥+5 𝑥−3 1−𝑥 2𝑥+1 𝑔 𝑥 = 𝐺 𝑥 = 𝑥+4 𝑥−5 3𝑥 −3𝑥+1 𝑔 𝑥 = 𝐺 𝑥 = 𝑥−3 5−𝑥  FIND OUT: how to graph 𝑝(𝑥) rational functions 𝑟 𝑥 = 𝑞(𝑥) if 𝑝(𝑥) is a constant and 𝑞(𝑥) is 2nd degree if 𝑝(𝑥) is a 1st degree and 𝑞(𝑥) is 2nd degree if 𝑝(𝑥) is a 2nd degree and 𝑞(𝑥) is 1st degree