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Worksheet 2.6A, Rational functions MATH 1410 (SOLUTIONS)
For each of the rational functions given below, do the following:
- Find the domain of the rational function.
- Reduce the rational function to lowest terms, if possible.
- Find the x- and y-intercepts of the graph of the rational function, if they exist.
- Determine the location of any vertical asymptotes or holes in the graph, if they exist.
- Analyze the end behavior of the rational function. Find the horizontal or slant asymptote, if one exists.
- Use a sign diagram and plot additional points, as needed, to sketch the graph of the rational function.
???
- a(x) = 2 x^2 โ 9 x^2 โ 9
- b(x) = x x โ 1
- c(x) = x^ + 3 x โ 2
- d(x) = (x + 1)(2x โ 2) (x โ 3)(x + 4)
- e(x) = (2x โ 1)(x + 2) (2x + 3)(3x โ 4)
- f (x) = x^2 โ 1 x^2 + x โ 6
- g(x) = x^2 โ 4 3 x^2 + x โ 4
- h(x) = x^2 โ 6 x + 8 x^2 โ x โ 12
- i(x) = x^2 โ 9 x^3 โ 4 x
- j(x) = 2 x + 1 x^2 + x + 1 Solutions.
- a(x) =^2 x
x^2 โ 9 has domain all real numbers except ยฑ3 so, in interval notation, the domain is
(โโ, โ3) โช (โ 3 , 3) โช (3, โ).
The rational function has zeroes when x = ยฑ
Its y-intercept occurs when y = 1. Vertical asymptotes are x = 3 and x = โ 3. y = 2 is a horizontal asymptote. Here is a graph of the curve, along with the two vertical asymptotes:
- b(x) = x x โ 1 has domain all real numbers except 1 so, in interval notation, the domain is
(โโ, 1) โช (1, โ).
The rational function intersects the axes at the origin. It has a vertical asymptote x = 1 and y = 1 is a horizontal asymptote. Here is a graph of the curve, along with the one vertical asymptote:
- e(x) = (2x^ โ^ 1)(x^ + 2) (2x + 3)(3x โ 4) has domain all real numbers except โ 32 , 43 so, in interval notation, the domain is (โโ, โ
The rational function has zeroes when x =
, x = โ 2.
Its y-intercept occurs when y =
Vertical asymptotes are x = โ
and x =
y =
is a horizontal asymptote. Here is a graph of the curve, along with the two vertical asymptotes:
- f (x) = x^2 โ 1 x^2 + x โ 6 has domain all real numbers except โ 3 , 2 so, in interval notation, the domain is
(โโ, โ3) โช (โ 3 , 2) โช (2, โ).
The rational function has zeroes when x = ยฑ 1. Its y-intercept occurs when y =
Vertical asymptotes are x = 2 and x = โ 3. y = 1 is a horizontal asymptote. Here is a graph of the curve, along with the two vertical asymptotes:
- g(x) = x^2 โ 4 3 x^2 + x โ 4 has domain all real numbers except โ 43 , 1 so, in interval notation, the domain is (โโ, โ
The rational function has zeroes when x = ยฑ 2. Its y-intercept occurs when y = 1. Vertical asymptotes are x = 1 and x = โ
y =
is a horizontal asymptote. Here is a graph of the curve, along with the two vertical asymptotes:
