# asymptote and domain of rational function

1. Write an equation that expresses the relationship. Use k as the constant of variation. d varies directly as the square of y.

2. Find the horizontal asymptote, if any, of the graph of the rational function. f(x) = 8x / (2x^2 + 1)

3. Find the domain of the rational function: g(x) = 6x / [(x-1)(x+7)]

4. Find the vertical asymptotes, if any, of the graph of the rational function: f(x) = x/(x^2 +4)

5. How many vertical asymptotes are possible for the rational function R(x)?

6. Find the domain of the rational function. g(x) = (x+ 4)/(x^2 - 4)

7. Choose the one alternative that best completes the statement or answers the equation.

Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval notation. 5x^2 - 3x >=8

8. Find the indicated intercepts of the graph of the function

9. Find the vertical asymptotes, if any, of the graph of the rational function: f(x) = x/[x(x-2)]

10. Let R(x) = (x^3 - 2x^2 - 7)/(x^2 + 1), what kinds of asymptotes does R(x) possess?

11. Use the graph of the rational function shown to complete the statement. As x-->2-, f(x)-->

12. Find the horizontal asymptote, if any, of the graph of the rational function: h(x) = 6x^3/(3x^2 + 1)

13. what is the equation of the slant asymptote for R(x) = (x^2 - x + 17) /(x^2 + 1)?

14. If R(x) is a rational function, then R(x) < 0 describes what?

15. Is there y-axis symmetry for the rational function f(x) = -4x^2 /(6x^4 - 3)?

16. Find the indicated intercept of the graph of the function: x-intercepts of f(x) = (x - 4)/(x^2 + 8x - 2)

17. is there origin symmetry for the rational function f(x) = 4x / (6x^2 + 1)?

18.y varies jointly as a and b and inversely as the square root of c. y = 48 when a = 4, b = 8, and c = 36. Find y when a = 2, b = 7, and c = 16.

19. solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation. (x + 9) /(x + 1) > 0

20. Find the horizontal asymptote, if any, of the graph of the rational function: g(x) = 6x^2 /(2x^2 + 1)

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#### Solution Summary

The solution provides detailed steps of finding the domain, intercepts, and asymptotes of the rational function.

Graphing of Function With Vertical & Oblique Asymptotes

1. When preparing to graph the rational function

y(x), this algebra is done.

(a) Specify, with reasons, the (largest) domain

of y(x). (You needn't repeat algebra)

(b) Find where the graph crosses the y-axis.

y(x) = x^4 - 2x^3 - 3x^2 - 3x â?" 1

x^3 - 3x^2 + x - 3

= x + 1 -(x - 1)(x + 2)

(x - 3)(x^2 + 1)

= x + 1 â?" 1/x â?" 3 â?" 1/x^2 + 1

(c) Give all vertical asymptotes, and the graph's behaviour near such asymptotes.

(d) Specify the equation of the oblique asymptote to the graph, and the co-ordinates of

all points, if any, where the graph crosses this asymptote.

(e) Specify the intervals of the x-axis, corresponding to values of y(x) which are greater

than the corresponding point on the oblique asymptote.