Rational Functions

4. Can a graph of a rational function have no vertical asymptote? If so, how?

5. Can a graph of a rational function have no x-intercepts? If so, how?

For the following exercises, find the domain of the rational functions.

7. [latex]f\left(x\right)=\frac{x+1}{{x}^{2}-1}[/latex]

9. [latex]f\left(x\right)=\frac{{x}^{2}+4x - 3}{{x}^{4}-5{x}^{2}+4}[/latex]

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

13. [latex]f\left(x\right)=\frac{x}{{x}^{2}+5x - 36}[/latex]

15. [latex]f\left(x\right)=\frac{3x - 4}{{x}^{3}-16x}[/latex]

For the following exercises, find the x– and y-intercepts for the functions.

21. [latex]f\left(x\right)=\frac{x}{{x}^{2}-x}[/latex]

23. [latex]f\left(x\right)=\frac{{x}^{2}+x+6}{{x}^{2}-10x+24}[/latex]

For the following exercises, describe the local and end behavior of the functions.

25. [latex]f\left(x\right)=\frac{x}{2x+1}[/latex]

27. [latex]f\left(x\right)=\frac{-2x}{x - 6}[/latex]

For the following exercises, find the slant asymptote of the functions.

31. [latex]f\left(x\right)=\frac{4{x}^{2}-10}{2x - 4}[/latex]

33. [latex]f\left(x\right)=\frac{6{x}^{3}-5x}{3{x}^{2}+4}[/latex]

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.

39. [latex]p\left(x\right)=\frac{2x - 3}{x+4}[/latex]

41. [latex]s\left(x\right)=\frac{4}{{\left(x - 2\right)}^{2}}[/latex]

43. [latex]f\left(x\right)=\frac{3{x}^{2}-14x - 5}{3{x}^{2}+8x - 16}[/latex]

45. [latex]a\left(x\right)=\frac{{x}^{2}+2x - 3}{{x}^{2}-1}[/latex]

47. [latex]h\left(x\right)=\frac{2{x}^{2}+ x - 1}{x - 4}[/latex]

For the following exercises, write an equation for a rational function with the given characteristics.

51. Vertical asymptotes at x = 5 and x = –5, x-intercepts at [latex]\left(2,0\right)[/latex] and [latex]\left(-1,0\right)[/latex], y-intercept at [latex]\left(0,4\right)[/latex]

53. Vertical asymptotes at [latex]x=-4[/latex] and [latex]x=-5[/latex], x-intercepts at [latex]\left(4,0\right)[/latex] and [latex]\left(-6,0\right)[/latex], Horizontal asymptote at [latex]y=7[/latex]

For the following exercises, use the graphs to write an equation for the function.

57.

63.

For the following exercises, use a calculator to graph [latex]f\left(x\right)[/latex]. Use the graph to solve [latex]f\left(x\right)>0[/latex].

71. [latex]f\left(x\right)=\frac{4}{2x - 3}[/latex]

73. [latex]f\left(x\right)=\frac{x+2}{\left(x - 1\right)\left(x - 4\right)}[/latex]

For the following exercises, identify the removable discontinuity.

75. [latex]f\left(x\right)=\frac{{x}^{2}-4}{x - 2}[/latex]

77. [latex]f\left(x\right)=\frac{{x}^{2}+x - 6}{x - 2}[/latex]

79. [latex]f\left(x\right)=\frac{{x}^{3}+{x}^{2}}{x+1}[/latex]

For the following exercises, express a rational function that describes the situation.

81. A large mixing tank currently contains 300 gallons of water, into which 8 pounds of sugar have been mixed. A tap will open, pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after t minutes.

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.

85. A rectangular box with a square base is to have a volume of 20 cubic feet. The material for the base costs 30 cents/ square foot. The material for the sides costs 10 cents/square foot. The material for the top costs 20 cents/square foot. Determine the dimensions that will yield minimum cost. Let x = length of the side of the base.

Modeling Using Variation

For the following exercises, write an equation describing the relationship of the given variables.

5. y varies directly as the square of x and when x = 4, y = 80

7. y varies directly as the cube of x and when x = 36, y = 24.

11. y varies inversely as the square of x and when x = 3, y = 2.

15. y varies inversely as the cube root of x and when x = 64, y = 5.

17. y varies jointly as x, z, and w and when x = 1, z = 2, w = 5, then y = 100.

19. y varies jointly as x and the square root of z and when x = 2 and z = 25, then y = 100.

21. y varies jointly as x and z and inversely as w. When x = 3, z = 5, and w = 6, then y = 10.

For the following exercises, use the given information to find the unknown value.

25. y varies directly as the square of x. When x = 2, then y = 16. Find y when x = 8.

29. y varies inversely with x. When x = 3, then y = 2. Find y when x = 1.

35. y varies jointly as x, z, and w. When x = 2, z = 1, and w = 12, then y = 72. Find y when x = 1, z = 2, and w = 3.

39. y varies jointly as the square of x and the cube of z and inversely as the square root of w. When x = 2, z = 2, and w = 64, then y = 12. Find y when x = 1, z = 3, and w = 4.

For the following exercises, use the given information to answer the questions.

51. The distance s that an object falls varies directly with the square of the time, t, of the fall. If an object falls 16 feet in one second, how long for it to fall 144 feet?

53. The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?