Multiplying and Dividing Rational Functions
Simplify: (5x+15)/(x+3)
5
Subtract: (x + 12) / (2x - 5) - (3x - 2) / (2x - 5)
(-2x + 14) / (2x - 5)
Solve: 1 / x - (x - 2) / 3x = 4 / (3x)
x = 1
What is the Vertical Asymptote of the function: (2x - 3)/(x - 4)?
x = 4
What is the difference between a closed and open circle?
A closed circle is included on a graph (and in the domain) and an open circle is NOT included in a graph or domain.
Multiply: (a^2 + 7a + 12)/(a^2 - 9) * (a - 3)/(a+ 3)
(a + 4)/(a - 3)
(2x^2 + 7x + 4) / (x - 4)(x + 3)
Solve: (x -2)/(x - 1) = x + 2
x = 0
What is the Horizontal Asymptote of the function: (2x - 3)/(x - 4)?
y = 2
How do you find the x and y intercepts?
You plug in 0 for x to find the y-intercept and set the numerator equal to zero to find the x-intercept.
Divide: (c^2 - c - 12)/(c^2 + 4c + 3) and (c^2 - 6c + 8)/(c^2 + 5c + 4)
(c + 4)/(c - 2)
Subtract: (x)/(x-4) - (6x)/(x - 5)
(-5x^2 + 19x)/(x - 4)(x - 5)
Solve: 56 / (x^2 - 2x - 15) - 6 / (x + 3) = 7 / (x - 5)
No solution
(x = 5 is extraneous)
Find the coordinate of hole in the function: (x^2 - 64)/(x^2 + 9x + 8)
(-8, 16/7)
What is a Removable Discontinuity (hole) and how do you find it?
After factoring the rational function, if there are any factors that cancel from the numerator AND denominator, set the factor equal to zero and that is a hole (removable discontinuity) on the graph.
Multiply: (x^2-25) / (4x-20) * 1 / (x^2 + 6x + 5)
1 / (4(x+1))
Add: 1 / (12xy^2) + 3y / (10x^2)
(5x + 18y^3) / 60x^2y^2)
Solve: (2x - 9) / (x - 7) + x / 2 = 5 / (x - 7)
x = -4
(x = 7 is extraneous)
Identify the x and y intercepts in the function: (x - 4)/(3x - 8)
x-intercept: (4, 0)
y-intercept: (0, 1/2)
How do you find the Horizontal and Vertical Asymptote?
To find Horizontal Asymptotes: compare the degree of the numerator and denominator. To find the Vertical Asymptotes: set the denominator equal to zero and solve
Solve: (3x + 6)/(x + 5) / (x + 2)/(x^2 - 25)
3(x - 5)
3x^2 / (16x^2 - 1) + 2 / (4x + 1) - x / (4x - 1)
(-x^2 + 7x - 2) / (4x + 1)(4x - 1)
Solve: (x^2 - 4)/(x + 3) + (x - 2)/(x + 3) = 2
x = 4
(x = -3 is extraneous)
Find the horizontal AND vertical asymptotes of the function (2x - 3)/(x - 4)
VA: x = 4
HA: y = 2
How do we find the domain and range of a rational function?
Exclude the values of the vertical asymptotes to find domain, and exclude the values of horizontal asymptotes to find range (also look at graph to confirm).